Past Combinatorial Theory Seminar

The study of counting point-hyperplane incidences in the $d$-dimensional space was initiated in the 1990's by Chazelle and became one of the central problems in discrete geometry. It has interesting connections to many other topics, such as additive combinatorics and theoretical computer science. Assuming a standard non-degeneracy condition, i.e., that no $s$ points are contained in the intersection of $s$ hyperplanes, the currently best known upper bound on the number of incidences of $m$ points and $n$ hyperplanes in $\mathbb{R}^d$ is $O((mn)^{1-1/(d+1)}+m+n)$. This bound by Apfelbaum and Sharir is based on geometrical space partitioning techniques, which apply only over the real numbers.

In this talk, we discuss a novel combinatorial approach to study such incidence problems over arbitrary fields. Perhaps surprisingly, this approach matches the best known bounds for point-hyperplane incidences in $\mathbb{R}^d$ for many interesting values of $m, n, d$. Moreover, in finite fields our bounds are sharp as a function of $m$ and $n$ in every dimension. This approach can also be used to study point-variety incidences and unit-distance problem in finite fields, giving tight bounds for both problems under a similar non-degeneracy assumption. Joint work with A. Milojevic and I. Tomon.

An interesting twist on classical subgraph containment problems in graph theory is the following: given a graph $H$ and a collection $\{G_1, \dots , G_m\}$ of graphs on a common vertex set $[n]$, what conditions on $G_i$ guarantee a copy of $H$ using at most one edge from each $G_i$? Such a subgraph is called transversal, and the above problem is closely related to the study of temporal graphs in Network Theory. In 2020 Joos and Kim showed that if $\delta(G_i)\geq n/2$, the collection contains a transversal Hamilton cycle. We improve on their result by showing that it actually contains every transversal Hamilton cycle if $\delta(G_i)\geq (1/2+o(1))n$. That is, for every function $\chi:[n]\to[m]$, there is a Hamilton cycle whose $i$-th edge belongs to $G_{\chi(i)}$.

This is joint work with Candida Bowtell, Patrick Morris and Katherine Staden.

The organizing principle of Ramsey theory is that in large mathematical structures, there are relatively large substructures which are homogeneous. This is quantified in combinatorics by the notion of Ramsey numbers $r(s,t)$, which denote the minimum $N$ such that in any red-blue coloring of the edges of the complete graph on $N$ vertices, there exists a red complete graph on $s$ vertices or a blue complete graph on $t$ vertices.

While the study of Ramsey numbers goes back almost one hundred years, to early papers of Ramsey and Erdős and Szekeres, the long-standing conjecture of Erdős that $r(s,t)$ has order of magnitude close to $t^{s-1}$ as $t \to \infty$ remains open in general. It took roughly sixty years before the order of magnitude of $r(3,t)$ was determined by Jeong Han Kim, who showed $r(3,t)$ has order of magnitude $t^2/\log t$ as $t \to \infty$. In this talk, we discuss a variety of new techniques, including the modern method of containers, which lead to a proof of the conjecture of Erdős that $r(4,t)$ is of order close to $t^3$.

One of the salient philosophies in our approach is that good Ramsey graphs hide inside pseudorandom graphs, and the long-standing emphasis of tackling Ramsey theory from the point of view of purely random graphs is superseded by pseudorandom graphs. Via these methods, we also come close to determining the well-studied related quantities known as Erdős-Rogers functions and discuss related hypergraph coloring problems and applications.

Joint work in part with Sam Mattheus, Dhruv Mubayi and David Conlon.

Models for networks that evolve and change over time are ubiquitous in a host of domains including modeling social networks, understanding the evolution of systems in proteomics, the study of the growth and spread of epidemics etc.

This talk will give a brief summary of three recent findings in this area where stochastic approximation techniques play an important role:

1. Understanding the effect and detectability of change point in the evolution of the system dynamics.
2. Reconstructing the initial "seed" that gave rise to the current network, sometimes referred to as Network Archeology.
3. The disparity in the behavior of different centrality measures such as degree and page rank centrality for measuring popularity in settings where there are vertices of different types such as majorities and minorities as well as insight analyzing such problems give for at first sight unrelated issues such as sampling rare groups within the network.

The main goal will to be convey unexpected findings in each of these three areas and in particular the "unreasonable effectiveness" of continuous time branching processes.

Let $G$ be a $d$-regular graph of growing degree on $n$ vertices. Form a random subgraph $G_p$ of $G$ by retaining edge of $G$ independently with probability $p=p(d)$. Which conditions on $G$ suffice to observe a phase transition at $p=1/d$, similar to that in the binomial random graph $G(n,p)$, or, say, in a random subgraph of the binary hypercube $Q^d$?

We argue that in the supercritical regime $p=(1+\epsilon)/d$, $\epsilon>0$ a small constant, postulating that every vertex subset $S$ of $G$ of at most $n/2$ vertices has its edge boundary at least $C|S|$, for some large enough constant $C=C(\epsilon)>0$, suffices to guarantee likely appearance of the giant component in $G_p$. Moreover, its asymptotic order is equal to that in the random graph $G(n,(1+\epsilon)/n)$, and all other components are typically much smaller.

We also give examples demonstrating tightness of our main result in several key senses.

A joint work with Sahar Diskin, Joshua Erde and Mihyun Kang.

A cluster graph is a disjoint union of complete graphs. We consider the random $G(n,p)$ graph on $n$ vertices with connection probability $p$, conditioned on the rare event of being a cluster graph. There are three main motivations for our study.

1. For $p = 1/2$, each random cluster graph occurs with the same probability, resulting in the uniform distribution over set partitions. Interpreting such a partition as a graph adds additional structural information.
2. To study how the law of a well-studied object like $G(n,p)$ changes when conditioned on a rare event; an evidence of this fact is that the conditioned random graph overcomes a phase transition at $p=1/2$ (not present in the dense $G(n,p)$ model).
3. The original motivation was an application to community detection. Taking a random cluster graph as a model for a prior distribution of a partition into communities leads to significantly better community-detection performance.

This is joint work with Martijn Gösgens, Lukas Lüchtrath, Elena Magnanini and Élie de Panafieu.

Coboundary expansion is a notion introduced by Linial and Meshulam, and by Gromov that combines combinatorics topology and linear algebra. Kaufman and Lubotzky observed its relation to "Property testing", and in recent years it has found several applications in theoretical computer science, including for error correcting codes (both classical and quantum), for PCP agreement tests, and even for studying polarization in social networks.

In the talk I will introduce this notion and some of its applications. No prior knowledge is assumed, of course.

I will talk about the behaviour of random maps on surfaces (for example, random triangulations) of given genus, when their size tends to infinity. Such questions can be asked from the viewpoint of the local behaviour (Benjamini-Schramm convergence) or global behaviour (diameter, Gromov Hausdorff convergence), and in both cases, much combinatorics is involved. I will survey the landmark results for the case of fixed genus, and state very recent results in which we manage to address the "high genus" regime, when the genus grows proportionally to the size – for this regime we establish isoperimetric inequalities and prove the long-suspected fact that the diameter is logarithmic with high probability.

Based on joint work with Thomas Budzinski and Baptiste Louf.

The saturation number of a graph is a famous and well-studied counterpoint to the Turán number, and the rainbow saturation number is a generalisation of the saturation number to the setting of coloured graphs. Specifically, for a given graph $F$, an edge-coloured graph is $F$-rainbow saturated if it does not contain a rainbow copy of $F$, but the addition of any non-edge in any colour creates a rainbow copy of $F$. The rainbow saturation number of $F$ is the minimum number of edges in an $F$-rainbow saturated graph on $n$ vertices. Girão, Lewis, and Popielarz conjectured that, like the saturation number, for all $F$ the rainbow saturation number is linear in $n$. I will present our attractive and elementary proof of this conjecture, and finish with a discussion of related results and open questions.

We show that there exists $C>1$, such that if $A$ is a subset of a non-alternating finite simple group $G$ of density $|A|/|G|= \alpha$, then $AA^{-1}AA^{-1}$ contains a subgroup of density at least $\alpha^{C}$. We will also give a corresponding (slightly weaker) statement for alternating groups.

To prove our results we introduce new hypercontractive inequalities for simple groups. These allow us to show that the (non-abelian) Fourier spectrum of indicators of 'global' sets are concentrated on the high-dimensional irreducible representations. Here globalness is a pseudorandomness notion reminiscent of the notion of spreadness.

The talk is based on joint works with David Ellis, Shai Evra, Guy Kindler, Nathan Lindzey, and Peter Keevash, and Dor Minzer. No prior knowledge of representation theory will be assumed.

A colouring rule is a way to colour the points $x$ of a probability space according to the colours of finitely many measure preserving tranformations of $x$. The rule is paradoxical if the rule can be satisfied a.e. by some colourings, but by none whose inverse images are measurable with respect to any finitely additive extension for which the transformations remain measure preserving. We show that proper graph colouring as a rule can be paradoxical. And we demonstrate rules defined via optimisation that are paradoxical. A connection to measure theoretic paradoxes is established.

The positive discrepancy of a graph $G$ is the maximum surplus of edges in an induced subgraph of $G$ compared to its expected size. This quantity is closely related to other well studied parameters, such as the minimum bisection and the spectral gap. I will talk about the extremal behavior of the positive discrepancy among graphs with given number of vertices and average degree, uncovering a surprising pattern. This leads to an almost complete solution of a problem of Alon on the minimum bisection and let's us extend the Alon-Boppana bound on the second eigenvalue to dense graphs.

Joint work with Eero Räty and Benny Sudakov.

The directed landscape is a random directed metric on the plane that is the scaling limit for models in the KPZ universality class (i.e. last passage percolation on $\mathbb{Z}^2$, TASEP). In this metric, typical pairs of points are connected by a unique geodesic.  However, certain exceptional pairs are connected by more exotic geodesic networks. The goal of this talk is to describe a full classification for these exceptional pairs.

An $n$-vertex graph $G$ is a $C$-expander if $|N(X)|\geq C|X|$ for every $X\subseteq V(G)$ with $|X|< n/2C$ and there is an edge between every two disjoint sets of at least $n/2C$ vertices.

We show that there is some constant $C>0$ for which every $C$-expander is Hamiltonian. In particular, this implies the well known conjecture of Krivelevich and Sudakov from 2003 on Hamilton cycles in $(n,d,\lambda)$-graphs. This completes a long line of research on the Hamiltonicity of sparse graphs, and has many applications.

Joint work with R. Montgomery, D. Munhá Correia, A. Pokrovskiy and B. Sudakov.

Brown-Erdős-Sós initiated the study of the maximum number of edges in an $n$-vertex $r$-graph such that no $k$ edges span at most $s$ vertices. If $s=rk-2k+2$ then this function is quadratic in $n$ and its asymptotic was previously known for $k=2,3,4$. I will present joint work with Stefan Glock, Jaehoon Kim, Lyuben Lichev and Shumin Sun where we resolve the cases $k=5,6,7$.

Given a matrix $A$ with integer entries, a subset $S$ of an abelian group and $r\in\mathbb N$, we say that $S$ is $(A,r)$-Rado if any $r$-colouring of $S$ yields a monochromatic solution to the system of equations $Ax=0$. A classical result of Rado characterises all those matrices $A$ such that $\mathbb N$ is $(A,r)$-Rado for all $r \in \mathbb N$. Rödl and Ruciński, and Friedgut, Rödl and Schacht proved a random version of Rado’s theorem where one considers a random subset of $[n]:=\{1,\dots,n\}$.

In this paper, we investigate the analogous random Ramsey problem in the more general setting of abelian groups. Given a sequence $(S_n)_{n\in\mathbb N}$ of finite subsets of abelian groups, let $S_{n,p}$ be a random subset of $S_n$ obtained by including each element of $S_n$ independently with probability $p$. We are interested in determining the probability threshold for $S_{n,p}$ being $(A,r)$-Rado.

Our main result is a general black box for hypergraphs which we use to tackle problems of this type. Using this tool in conjunction with a series of supersaturation results, we determine the probability threshold for a number of different cases. A consequence of the Green-Tao theorem is the van der Waerden theorem for the primes: every finite colouring of the primes contains arbitrarily long monochromatic arithmetic progressions. Using our machinery, we obtain a random version of this result. We also prove a novel supersaturation result for $[n]^d$ and use it to prove an integer lattice generalisation of the random version of Rado's theorem.

This is joint work with Andrea Freschi and Andrew Treglown (both University of Birmingham).

For integers $k \ge 1$ and $n \ge 2k+1$, the Kneser graph $K(n,k)$ has as vertices all $k$-element subsets of an $n$-element ground set, and an edge between any two disjoint sets. It has been conjectured since the 1970s that all Kneser graphs admit a Hamilton cycle, with one notable exception, namely the Petersen graph $K(5,2)$. This problem received considerable attention in the literature, including a recent solution for the sparsest case $n=2k+1$. The main contribution of our work is to prove the conjecture in full generality. We also extend this Hamiltonicity result to all connected generalized Johnson graphs (except the Petersen graph). The generalized Johnson graph $J(n,k,s)$ has as vertices all $k$-element subsets of an $n$-element ground set, and an edge between any two sets whose intersection has size exactly $s$. Clearly, we have $K(n,k)=J(n,k,0)$, i.e., generalized Johnson graphs include Kneser graphs as a special case. Our results imply that all known families of vertex-transitive graphs defined by intersecting set systems have a Hamilton cycle, which settles an interesting special case of Lovász' conjecture on Hamilton cycles in vertex-transitive graphs from 1970. Our main technical innovation is to study cycles in Kneser graphs by a kinetic system of multiple gliders that move at different speeds and that interact over time, reminiscent of the gliders in Conway’s Game of Life, and to analyze this system combinatorially and via linear algebra.

This is joint work with my students Arturo Merino (TU Berlin) and Namrata (Warwick).

Random temporal graphs are a version of the classical Erdős-Rényi random graph $G(n,p)$ where additionally, each edge has a distinct random time stamp, and connectivity is constrained to sequences of edges with increasing time stamps. We are interested in the asymptotics for the distances in such graphs, mostly in the regime of interest where the average degree $np$ is of order $\log n$ ('near' the phase transition).

More specifically, we will discuss the asymptotic lengths of increasing paths: the lengths of the shortest and longest paths between typical vertices, as well as the maxima between any two vertices; this also covers the (temporal) diameter. In the regime $np \gg \log n$, longest increasing paths were studied by Angel, Ferber, Sudakov and Tassion.

The talk contains joint work with Nicolas Broutin and Gábor Lugosi.

In this talk we will review the problem of sharpness in percolation, tracing its origins back to the seminal works of Menshikov, Grimmett-Marstrand and Aizenman-Barsky, which successfully settled the question in the context of Bernoulli independent percolation. Then we will present some recent advancements on the field, which have opened up the possibility of investigating dependent percolation models. Special emphasis will be given to the Interpolation technique, which has proved itself very effective. In particular, it has been used to establish the sharpness for Interlacements Percolation, a model introduced by Sznitman with notoriously intricate dependencies.

This talk is based on a joint work with Duminil-Copin, Goswami, Rodriguez and Severo

A rooted tree $T$ has degree sequence $(d_1,\ldots,d_n)$ if $T$ has vertex set $[n]$ and vertex $i$ has $d_i$ children for each $i$ in $[n]$. 

I will describe a line-breaking construction of random rooted trees with given degree sequences, as well as a way of coupling random trees with different degree sequences that also couples their heights to one another. 

The construction and the coupling have several consequences, and I'll try to explain some of these in the talk.

First, let $T$ be a branching process tree with critical—mean one—offspring distribution, and let $T_n$ have the law of $T$ conditioned to have size $n$. Then the following both hold.
1) $\operatorname{height}(T_n)/\log(n)$ tends to infinity in probability. 
2) If the offspring distribution has infinite variance then $\operatorname{height}(T_n)/n^{1/2}$ tends to $0$ in probability. This result settles a conjecture of Svante Janson.

The next two statements relate to random rooted trees with given degree sequences. 
1) For any $\varepsilon > 0$ there is $C > 0$ such that the following holds. If $T$ is a random tree with degree sequence $(d_1,\ldots,d_n)$ and at least $\varepsilon n$ leaves, then $\mathbb{E}(\operatorname{height}(T)) < C \sqrt{n}$. 
2) Consider any random tree $T$ with a fixed degree sequence such that $T$ has no vertices with exactly one child. Then $\operatorname{height}(T)$ is stochastically less than $\operatorname{height}(B)$, where $B$ is a random binary tree of the same size as $T$ (or size one greater, if $T$ has even size). 

Euclidean Ramsey Theory is a natural multidimensional version of Ramsey Theory. A subset of Euclidean space is called Ramsey if, for any $k$, whenever we partition Euclidean space of sufficiently high dimension into $k$ classes, one class much contain a congruent copy of our subset. It is still unknown which sets are Ramsey. We will discuss background on this and then proceed to some recent results.

In this talk, I will present new results addressing two rather well-known problems on the embeddability of planar graphs on point-sets in the plane. The first problem, often attributed to Mohar, asks for the asymptotics of the minimum size of so-called universal point sets, i.e. point sets that simultaneously allow straight-line embeddings of all planar graphs on $n$ vertices. In the first half of the talk I will present a family of point sets of size $O(n)$ that allow straight-line embeddings of a large family of $n$-vertex planar graphs, including all bipartite planar graphs. In the second half of the talk, I will present a family of $(3+o(1))\log_2(n)$ planar graphs on $n$ vertices that cannot be simultaneously embedded straight-line on a common set of $n$ points in the plane. This significantly strengthens the previously best known exponential bound.

I will talk about two separate projects where random walks are building a random tree, yielding preferential attachment behaviour from completely local mechanisms.
First, the Tree Builder Random Walk is a randomly growing tree, built by a walker as she is walking around the tree. At each time $n$, she adds a leaf to her current vertex with probability $n^{-\gamma}, \gamma\in(2/3, 1]$, then moves to a uniform random neighbor on the possibly modified tree. We show that the tree process at its growth times, after a random finite number of steps, can be coupled to be identical to the Barabási-Albert preferential attachment tree model. This coupling implies that many properties known for the BA-model, such as diameter and degree distribution, can be directly transferred to our model. Joint work with János Engländer, Giulio Iacobelli, and Rodrigo Ribeiro. Second, we introduce a network-of-networks model for physical networks: we randomly grow subgraphs of an ambient graph (say, a box of $\mathbb{Z}^d$) until they hit each other, building a tree from how these spatially extended nodes touch each other. We compute non-rigorously the degree distribution exponent of this tree in large generality, and provide a rigorous analysis when the nodes are loop-erased random walks in dimension $d=2$ or $d\geq 5$, using a connection with the Uniform Spanning Tree. Joint work with Ádám Timár, Sigurdur Örn Stefánsson, Ivan Bonamassa, and Márton Pósfai. (See https://arxiv.org/abs/2306.01583)

We show that given $m$ disjoint chains in the Boolean lattice, we can create $m$ disjoint skipless chains that cover the same elements (where we call a chain skipless if any two consecutive elements differ in size by exactly one). By using this result we are able to answer two conjectures about the asymptotics of induced saturation numbers for the antichain, which are defined as follows. For positive integers $k$ and $n$, a family $\mathcal{F}$ of subsets of $\{1,\dots,n\}$ is $k$-antichain saturated if it does not contain an antichain of size $k$ (as induced subposet), but adding any set to $\mathcal{F}$ creates an antichain of size $k$. We use $\textrm{sat}^{\ast}(n,k)$ to denote the smallest size of such a family. With more work we pinpoint the exact value of $\textrm{sat}^{\ast}(n,k)$, for all $k$ and sufficiently large $n$. Previously, exact values for $\textrm{sat}^{\ast}(n,k)$ were only known for $k$ up to 6. We also show that for any poset $\mathcal{P}$, its induced saturation number (which is defined similar as for the antichain) grows at most polynomially: $\textrm{sat}^{\ast}(n, \mathcal{P})=O(n^c)$, where $c \leq |\mathcal{P}|^2/4+1$. This is based on joint works with Carla Groenland, Maria-Romina Ivan, Hugo Jacob and Tom Johnston.

We determine the minimal forbidden minors characterising the class of countable graphs that embed into some compact surface. We will also discuss Thomas’s conjecture that the Robertson—Seymour Graph Minor Theorem extends to countably infinite graphs. [https://arxiv.org/abs/2301.11042]

Consider the following "controlled" random graph process: edges of the complete graph are revealed one by one in random order to an online algorithm, which immediately decides whether to retain each observed edge. The algorithm's objective is to construct a graph property within specified constraints on the total number of observed edges ("time") and the total number of retained edges ("budget").

During this talk, I will present results in this model for natural graph properties, such as connectivity, Hamiltonicity, and containment of fixed-size subgraphs. Specifically, I will describe a strategy to construct a Hamilton cycle at the hitting time for minimum degree 2 by retaining a linear number of edges. This extends the classical hitting time result for Hamiltonicity originally established by Ajtai–Komlós–Szemerédi and Bollobás.

The talk is based on joint work with Alan Frieze and Michael Krivelevich.

We show that every 2-coloring of the natural numbers and any finite coloring of the rationals contains monochromatic sets of the form $\{x, y, xy, x+y\}$. We also discuss generalizations and obstructions to extending this result to arbitrary finite coloring of the naturals. This is partially based on joint work with Marcin Sabok.

Motivated by the desire to construct large independent sets in random graphs, Karp and Sipser modified the usual greedy construction to yield an algorithm that outputs an independent set with a large cardinal called the Karp-Sipser core. When run on the Erdős-Rényi $G(n,c/n)$ random graph, this algorithm is optimal as long as $c < e$. We will present the proof of a physics conjecture of Bauer and Golinelli (2002) stating that at criticality, the size of the Karp-Sipser core is of order $n^{3/5}$. Along the way we shall highlight the similarities and differences with the usual greedy algorithm and the $k$-core algorithm.
Based on a joint work with Nicolas Curien and Thomas Budzinski.

A $k$-block in a graph is a set of $k$ vertices every two of which are joined by $k$ vertex disjoint paths. By a result of Weissauer, graphs with no $k$-blocks admit tree-decompositions with especially useful structure. While several constructions show that it is probably very difficult to characterize induced subgraph obstructions for bounded tree width, a lot can be said about graphs with no $k$-blocks. On the other hand, forbidding induced subgraphs places significant restrictions on the structure of a $k$-block in a graphs. We will discuss this phenomenon and its consequences in the study of tree-decompositions in classes of graphs defined by forbidden induced subgraphs.

Let $G$ be a $d$-regular graph of growing degree on $n$ vertices, and form a random subgraph $G_p$ of $G$ by retaining edge of $G$ independently with probability $p=p(d)$. Which conditions on $G$ suffice to observe a phase transition at $p=1/d$, similar to that in the binomial random graph $G(n,p)$, or, say, in a random subgraph of the binary hypercube $Q^d$?

We argue that in the supercritical regime $p=(1+\epsilon)/d$, $\epsilon>0$ being a small constant, postulating that every vertex subset $S$ of $G$ of at most $n/2$ vertices has its edge boundary at least $C|S|$, for some large enough constant $C=C(\epsilon)>0$, suffices to guarantee the likely appearance of the giant component in $G_p$. Moreover, its asymptotic order is equal to that in the random graph $G(n,(1+\epsilon)/n)$, and all other components are typically much smaller.

We further give examples demonstrating the tightness of this result in several key senses.

A joint work with Sahar Diskin, Joshua Erde and Mihyun Kang.

A factorisation $x=u_1u_2\cdots$ of an infinite word $x$ on alphabet $X$ is called ‘super-monochromatic’, for a given colouring of the finite words $X^{\ast}$ on alphabet $X$, if each word $u_{k_1}u_{k_2}\cdots u_{k_n}$, where $k_1<\cdots<k_n$, is the same colour. A direct application of Hindman’s theorem shows that if $x$ is eventually periodic, then for every finite colouring of $X^{\ast}$, there exist a suffix of $x$ that admits a super-monochromatic factorisation. What about the converse?

In this talk we show that the converse does indeed hold: thus a word $x$ is eventually periodic if and only if for every finite colouring of $X^{\ast}$ there is a suffix of $x$ having a super-monochromatic factorisation. This has been a conjecture in the community for some time. Our main tool is a Ramsey result about alternating sums. This provides a strong link between Ramsey theory and the combinatorics of infinite words.

Joint work with Imre Leader and Luca Q. Zamboni

In the critical beta-splitting model of a random $n$-leaf rooted tree, clades (subtrees) are recursively split into sub-clades, and a clade of $m$ leaves is split into sub-clades containing $i$ and $m-i$ leaves with probabilities $\propto 1/(i(m-i))$. This model turns out to have interesting properties. There is a canonical embedding into a continuous-time model ($\operatorname{CTCS}(n)$). There is an inductive construction of $\operatorname{CTCS}(n)$ as $n$ increases, analogous to the stick-breaking constructions of the uniform random tree and its limit continuum random tree. We study the heights of leaves and the limit fringe distribution relative to a random leaf. In addition to familiar probabilistic methods, there are analytic methods (developed by co-author Boris Pittel), based on explicit recurrences, which often give more precise results. So this model provides an interesting concrete setting in which to compare and contrast these methods. Many open problems remain.
Preprints at https://arxiv.org/abs/2302.05066 and https://arxiv.org/abs/2303.02529

More than 30 year ago Jerrum studied the planted clique problem and proved that under worst-case initialization Metropolis fails to recover planted cliques of size $\ll n^{1/2}$ in the Erdős-Rényi graph $G(n,1/2)$. This result is classically cited in the literature of the problem, as the "first evidence" that finding planted cliques of size much smaller than square root $n$ is "algorithmically hard". Cliques of size $\gg n^{1/2}$ are easy to find using simple algorithms. In a recent work we show that the Metropolis process actually fails to find planted cliques under worst-case initialization for cliques up to size almost linear in $n$. Thus the algorithm fails well beyond the $\sqrt{n}$ "conjectured algorithmic threshold". We also prove that, for a large parameter regime, that the Metropolis process fails also under "natural initialization". Our results resolve some open questions posed by Jerrum in 1992. Based on joint work with Zongchen Chen and Iias Zadik.

Magnant and Martin conjectured that every $d$-regular graph on $n$ vertices can be covered by $n/(d+1)$ vertex-disjoint paths. Gruslys and Letzter verified this conjecture in the dense case, even for cycles rather than paths. We prove the analogous result for directed graphs and oriented graphs, that is, for all $\alpha>0$, there exists $n_0=n_0(\alpha)$ such that every $d$-regular digraph on $n$ vertices with $d \ge \alpha n $ can be covered by at most $n/(d+1)$ vertex-disjoint cycles. Moreover if $G$ is an oriented graph, then $n/(2d+1)$ cycles suffice. This also establishes Jackson's long standing conjecture for large $n$ that every $d$-regular oriented graph on $n$ vertices with $n\leq 4d+1$ is Hamiltonian.
This is joint work with Viresh Patel and  Mehmet Akif Yıldız.

The speakers are Maya Stein (University of Chile), Mathias Schacht (Hamburg), János Pach (Rényi Institute, Hungary and IST Austria), Marthe Bonamy (Bordeaux), Mehtaab Sawhney (Cambridge/MIT), and Julian Sahasrabudhe (Cambridge). Please see the event website for further details including titles, abstracts, and timings. Anyone interested is welcome to attend, and no registration is required.

Let $p_c$ and $q_c$ be the threshold and the expectation threshold, respectively, of an increasing family $F$ of subsets of a finite set $X$. Recently, Park and Pham proved Kahn–Kalai conjecture stating that a not-too-large multiple of $q_c$ is an upper bound on $p_c$. In the talk, I will present a slight improvement to the Park–Pham theorem, which is obtained from transferring the threshold result from the small $p$ regime to general $p$. Based on joint work with Oliver Riordan.

"Colouring" a tournament means partitioning its vertex set into acylic subsets; and the "domination number" is the size of the smallest set of vertices with no common in-neighbour. In some ways these are like the corresponding concepts for graphs, but in some ways they are very different. We give a survey of some recent results and open questions on these topics.

Joint with Tung Nguyen and Alex Scott.

We determine the minimal forbidden minors characterising the class of countable graphs that embed into some compact surface. We will also discuss Thomas’s conjecture that the Robertson—Seymour Graph Minor Theorem extends to countably infinite graphs. [https://arxiv.org/abs/2301.11042]

A classic result of Chvatál and Erdős (1972) asserts that, if the vertex-connectivity of a graph G is at least as large as its independence number, then G has a Hamilton cycle. We prove a similar result, implying that a graph G is pancyclic, namely it contains cycles of all lengths between 3 and |G|: we show that if |G| is large and the vertex-connectivity of G is larger than its independence number, then G is pancyclic. This confirms a conjecture of Jackson and Ordaz (1990) for large graphs.

I will discuss statistical inference problems on edge-correlated stochastic block models. We determine the information-theoretic threshold for exact recovery of the latent vertex correspondence between two correlated block models, a task known as graph matching. As an application, we show how one can exactly recover the latent communities using multiple correlated graphs in parameter regimes where it is information-theoretically impossible to do so using just a single graph. Furthermore, we obtain the precise threshold for exact community recovery using multiple correlated graphs, which captures the interplay between the community recovery and graph matching tasks. This is based on joint work with Julia Gaudio and Anirudh Sridhar.

In 1977, Erdős and Hajnal made the conjecture that, for every graph $H$, there exists $c>0$ such that every $H$-free graph $G$ has a clique or stable set of size at least $|G|^c$; and they proved that this is true with $|G|^c$ replaced by $2^{c\sqrt{\log |G|}}$. Until now, there has been no improvement on this result (for general $H$). We recently proved a strengthening: that for every graph $H$, there exists $c>0$ such that every $H$-free graph $G$ with $|G|\ge 2$ has a clique or stable set of size at least $2^{c\sqrt{\log |G| \log\log|G|}}$. This talk will outline the proof. Joint work with Matija Bucić, Tung Nguyen and Alex Scott.

Random greedy algorithms became ubiquitous in Combinatorics after Rödl's nibble (semi-random method), which was repeatedly refined for various applications, such as iterative graph colouring algorithms (Molloy-Reed) and lower bounds for the Ramsey number $R(3,t)$ via the triangle-free process (Bohman-Keevash / Fiz Pontiveros-Griffiths-Morris). More recently, when combined with absorption, they have played a key role in many existence and approximate counting results for combinatorial structures, following a paradigm established by my proofs of the Existence of Designs and Wilson's Conjecture on the number of Steiner Triple Systems. Here absorption (converting approximate solutions to exact solutions) is generally the most challenging task, which has spurred the development of many new ideas, including my Randomised Algebraic Construction method, the Kühn-Osthus Iterative Absorption method and Montgomery's Addition Structures (for attacking the Ryser-Brualdi-Stein Conjecture). The design and analysis of a suitable guiding mechanism for the random process can also come with major challenges, such as in the recent proof of Erdős' Conjecture on Steiner Triple Systems of high girth (Kwan-Sah-Sawhney-Simkin). This talk will survey some of this background and also mention some recent results on the Queens Problem (Bowtell-Keevash / Luria-Simkin / Simkin) and the Existence of Subspace Designs (Keevash-Sah-Sawhney). I may also mention recent solutions of the Talagrand / Kahn-Kalai Threshold Conjectures (Frankston-Kahn-Narayanan-Park / Park-Pham) and thresholds for Steiner Triple Systems / Latin Squares (Keevash / Jain-Pham), where the key to my proof is constructing a suitable spread measure via a guided random process.

A regular bipartite tournament is an orientation of a complete balanced bipartite graph $K_{2n,2n}$ where every vertex has its in- and outdegree both equal to $n$. In 1981, Jackson conjectured that any regular bipartite tournament can be decomposed into Hamilton cycles. We prove this conjecture for sufficiently large $n$. Along the way, we also prove several further results, including a conjecture of Liebenau and Pehova on Hamilton decompositions of dense bipartite digraphs.

Let $p \ge 3$ be a prime integer. The density of a non-empty solution set of a system of affine equations $L_i(x) = b_i$, $i=1,\dots,k$ on a vector space over the field $\mathbb{F}_p$ is determined by the dimension of the linear subspace $\langle L_1,\dots,L_k \rangle$, and tends to $0$ with the dimension of that subspace. In particular, if the solution set is dense, then the system of equations contains at most boundedly many pairwise distinct linear forms. In the more general setting of systems of affine conditions $L_i(x) \in E_i$ for some strict subsets $E_i$ of $\mathbb{F}_p$ and where the solution set and its density are taken inside $S^n$ for some non-empty subset $S$ of $\mathbb{F}_p$ (such as $\{0,1\}$), we can however usually find subsets of $S^n$ with density at least $1/2$ but such that the linear subspace $\langle L_1,\dots,L_k \rangle$ has arbitrarily high dimension. We shall nonetheless establish an approximate boundedness result: if the solution set of a system of affine conditions is dense, then it contains the solution set of a system of boundedly many affine conditions and which has approximately the same density as the original solution set. Using a recent generalisation by Gowers and the speaker of a result of Green and Tao on the equidistribution of high-rank polynomials on finite prime fields we shall furthermore prove a weaker analogous result for polynomials of small degree.

Based on joint work with Timothy Gowers (College de France and University of Cambridge).

I will discuss the difficult problem of proving reasonable bounds in the multidimensional generalization of Szemerédi's theorem and describe a proof of such bounds for sets lacking nontrivial configurations of the form (x,y), (x,y+z), (x,y+2z), (x+z,y) in the finite field model setting.

In this talk, I will review some recent progress on random graph matching problems, that is, to recover the vertex correspondence between a pair of correlated random graphs from the observation of two unlabelled graphs. In this talk, I will touch issues of information threshold, efficient algorithms as well as complexity theory. This is based on joint works with Hang Du, Shuyang Gong and Zhangsong Li.

We present two ways of obtaining hypercontractive inequalities for low-degree functions on compact Lie groups: one based on Ricci curvature bounds, the Bakry-Emery criterion and the representation theory of compact Lie groups, and another based on a (very different) probabilistic coupling approach. As applications we make progress on a question of Gowers concerning product-free subsets of the special unitary groups, and we also obtain 'mixing' inequalities for the special unitary groups, the special orthogonal groups, the spin groups and the compact symplectic groups. We expect that the latter inequalities will have applications in physics.

Based on joint work with Guy Kindler (HUJI), Noam Lifshitz (HUJI) and Dor Minzer (MIT).

The triangle removal lemma of Ruzsa and Szemerédi is a fundamental result in extremal graph theory; very roughly speaking, it says that if a graph is "far" from triangle-free, then it contains "many" triangles. Despite decades of research, there is still a lot that we don't understand about this simple statement; for example, our understanding of the quantitative dependencies is very poor.

In this talk, I will discuss asymmetric versions of the triangle removal lemma, where in some cases we can get almost optimal quantitative bounds. The proofs use a mix of ideas coming from graph theory, number theory, probabilistic combinatorics, and Ramsey theory.

Based on joint work with Lior Gishboliner and Asaf Shapira.

We consider a bond percolation on the hypercube in the supercritical regime. We derive vertex-expansion properties of the giant component. As a consequence we obtain upper bounds on the diameter of the giant component and the mixing time of the lazy random walk on the giant component. This talk is based on joint work with Joshua Erde and Michael Krivelevich.