Past Combinatorial Theory Seminar

Graphs (and structures) which have a linear ordering of their vertices with given local properties have a rich spectrum of complexities. Some have full power of class NP (and thus no dichotomy) but for biconnected patterns we get dichotomy. This also displays the importance of Sparse Incomparability Lemma. This is a joint work with Gabor Kun (Budapest).

The class of $t$-perfect graphs consists of graphs whose stable set polytopes are defined by their non-negativity, edge inequalities, and odd circuit inequalities. These were first studied by Chvátal in 1975, motivated by the related and well-studied class of perfect graphs. While perfect graphs are easy to colour, the same is not true for $t$-perfect graphs; numerous questions and conjectures have been posed, and even the most basic, on whether there exists some $k$ such that every $t$-perfect graph is $k$-colourable, has remained open since 1994. I will talk about joint work with Maria Chudnovsky, Linda Cook, James Davies, and Sang-il Oum in which we establish the first finite bound and show that a little less than 200 000 colours suffice.

I will discuss various results and conjectures about off-diagonal hypergraph Ramsey numbers, focusing on recent developments.

A set in the Euclidean plane is called an integer distance set if the distance between any pair of its points is an integer.  All so-far-known integer distance sets have all but up to four of their points on a single line or circle; and it had long been suspected, going back to Erdős, that any integer distance set must be of this special form. In a recent work, joint with Marina Iliopoulou and Sarah Peluse, we developed a new approach to the problem, which enabled us to make the first progress towards confirming this suspicion.  In the talk, I will discuss the study of integer distance sets, its connections with other problems, and our new developments.

Upper bounds on the number of incidences between points and lines, tubes, and other geometric objects, have many applications in combinatorics and analysis. On the other hand, much less is known about lower bounds. We prove a general lower bound for the number of incidences between points and tubes in the plane under a natural spacing condition. In particular, if you take $n$ points in the unit square and draw a line through each point, then there is a non-trivial point-line pair with distance at most $n^{-2/3+o(1)}$. This quickly implies that any $n$ points in the unit square define a triangle of area at most $n^{-7/6+o(1)}$, giving a new upper bound for the Heilbronn's triangle problem.

Joint work with Alex Cohen and Cosmin Pohoata.

The normal covering number $\gamma(G)$ of a finite group $G$ is the minimal size of a collection of proper subgroups whose conjugates cover the group. This definition is motivated by number theory and related to the concept of intersective polynomials. For the symmetric and alternating groups we will see how these numbers are closely connected to some elementary (as in "relating to basic concepts", not "easy") problems in additive combinatorics, and we will use this connection to better understand the asymptotics of $\gamma(S_n)$ and $\gamma(A_n)$ as $n$ tends to infinity.

Paraphrasing the title of Riemann’s famous lecture of 1854 I ask: What is the most rudimentary notion of a geometry? A possible answer is a path system: Consider a finite set of “points” $x_1,…,x_n$ and provide a recipe how to walk between $x_i$ and $x_j$ for all $i\neq j$, namely decide on a path $P_{ij}$, i.e., a sequence of points that starts at $x_i$ and ends at $x_j$, where $P_{ji}$ is $P_{ij}$, in reverse order. The main property that we consider is consistency. A path system is called consistent if it is closed under taking subpaths. What do such systems look like? How to generate all of them? We still do not know. One way to generate a consistent path system is to associate a positive number $w_{ij}>0$ with every pair and let $P_{ij}$ be the corresponding $w$-shortest path between $x_i$ and $x_j$. Such a path system is called metrical. It turns out that the class of consistent path systems is way richer than the metrical ones.

My main emphasis in this lecture is on what we don’t know and wish to know, yet there is already a considerable body of work that we have done on the subject.

The new results that I will present are joint with my student Daniel Cizma as well as with him and with Maria Chudnovsky.

I'll discuss how a certain notion of symmetry captures the computational complexity of approximating homomorphism problems between relational structures, also known as constraint satisfaction problems. I'll present recent results on inapproximability of conflict-free and linearly-ordered hypergraph colourings and solvability of systems of equations.

In 1975, Bollobás, Erdős, and Szemerédi asked the following Zarankiewicz-type problem. What is the smallest $\tau$ such that an $n \times n \times n$ tripartite graph with minimum degree $n + \tau$ must contain $K_{t, t, t}$? They further conjectured that $\tau = O(n^{1/2})$ when $t = 2$.

I will discuss our proof that $\tau = O(n^{1 - 1/t})$ (confirming their conjecture) and an infinite family of extremal examples. The bound $O(n^{1 - 1/t})$ is best possible whenever the Kővári-Sós-Turán bound $\operatorname{ex}(n, K_{t, t}) = O(n^{2 - 1/t})$ is (which is widely-conjectured to be the case).

This is joint work with Francesco Di Braccio (LSE).

In a recent breakthrough Campos, Griffiths, Morris and Sahasrabudhe obtained the first exponential improvement of the upper bound on the classical Ramsey numbers since 1935. I will outline a reinterpretation of their proof, replacing the underlying book algorithm with a simple inductive statement. In particular, I will present a complete proof of an improved upper bound on the off-diagonal Ramsey numbers and describe the main steps involved in improving their upper bound for the diagonal Ramsey numbers to $R(k,k)\le(3.8)^k$ for sufficiently large $k$.

Based on joint work with Parth Gupta, Ndiame Ndiaye, and Louis Wei.

Let $(V(u))$ be a branching random walk and $(\eta(u))$ be i.i.d marks on the vertices. To each path $\xi$ of the tree, we associate the discounted sum $D(\xi)$ which is the sum of the $\exp(V(u))\eta_u$ along the path. We study conditions under which $\sup_\xi D(\xi)$ is finite, answering an open question of Aldous and Bandyopadhyay. We will see that this problem is related to the study of the local time process of the branching random walk along a path. It is a joint work with Yueyun Hu and Zhan Shi.

A zero-one matrix $M$ is said to contain another zero-one matrix $A$ if we can delete some rows and columns of $M$ and replace some 1-entries with 0-entries such that the resulting matrix is $A$. The extremal number of $A$, denoted $\operatorname{ex}(n,A)$, is the maximum number of 1-entries that an $n\times n$ zero-one matrix can have without containing $A$. The systematic study of this function for various patterns $A$ goes back to the work of Furedi and Hajnal from 1992, and the field has many connections to other areas of mathematics and theoretical computer science. The problem has been particularly extensively studied for so-called acyclic matrices, but very little is known about the general case (that is, the case where $A$ is not necessarily acyclic). We prove the first asymptotically tight general result by showing that if $A$ has at most $t$ 1-entries in every row, then $\operatorname{ex}(n,A)\leq n^{2-1/t+o(1)}$. This verifies a conjecture of Methuku and Tomon.

Our result also provides the first tight general bound for the extremal number of vertex-ordered graphs with interval chromatic number two, generalizing a celebrated result of Furedi, and Alon, Krivelevich and Sudakov about the (unordered) extremal number of bipartite graphs with maximum degree $t$ in one of the vertex classes.

Joint work with Barnabas Janzer, Van Magnan and Abhishek Methuku.

The celebrated Szemeredi-Trotter theorem states that the maximum number of incidences between $n$ points and $n$ lines in the plane is $\mathcal{O}(n^{4/3})$, which is asymptotically tight.

Solymosi conjectured that this bound drops to $o(n^{4/3})$ if we exclude subconfigurations isomorphic to any fixed point-line configuration. We describe a construction disproving this conjecture. On the other hand, we prove new upper bounds on the number of incidences for configurations that avoid certain subconfigurations. Joint work with Martin Balko.

In a graph $H$ whose edges are coloured (not necessarily properly) a rainbow copy of a graph $G$ is a (not necessarily induced) subgraph of $H$ that is isomorphic to $G$ and whose edges are all coloured differently. In this talk I will explain why the problem of finding such rainbow copies is interesting, survey what we know, concentrating mainly on the case where $G$ is a Hamilton cycle, and then tell you a bit about a new result about finding rainbow Hamilton cycles resiliently in random graphs (which is joint work with Peter Allen and Liana Yepremyan).

The classical lower-tail problem for triangles in random graphs asks the following: given $\eta\in[0,1)$, what is the probability that $G(n,p)$ contains at most $\eta$ times the expected number of triangles?  When $p=o(n^{-1/2})$ or $p = \omega(n^{-1/2})$ the asymptotics of the logarithm of this probability are known via Janson's inequality in the former case and regularity or container methods in the latter case.

We prove for the first time asymptotic formulas for the logarithm of the lower tail probability when $p=c n^{-1/2}$ for $c$ constant.  Our results apply for all $c$ when $\eta \ge 1/2$ and for $c$  small enough when $\eta < 1/2$.  For the special case $\eta=0$ of triangle-freeness, our results prove that a phase transition occurs as $c$ varies (in the sense of a non-analyticity of the rate function), while for $\eta \ge 1/2$ we prove that no phase transition occurs.

Our method involves ingredients from algorithms and statistical physics including rapid mixing of Markov chains and the cluster expansion.  We complement our asymptotic formulas with efficient algorithms to approximately sample from $G(n,p)$ conditioned on the lower tail event.

Joint work with Will Perkins, Aditya Potukuchi and Michael Simkin.

We give an exponential improvement on the upper bound for the $r$-colour diagonal Ramsey number for all $r$. The proof relies on geometric insights and offers a simplified proof in the case of $r=2$.

Joint Work with: Paul Ballister, Béla Bollobás, Marcelo Campos, Simon Griffiths, Rob Morris, Julian Sahasrabudhe and Marius Tiba.

We show that an $n$-vertex $k$-uniform hypergraph, where all $(k-1)$-subsets that are supported by an edge are in fact supported by at least $n/2+o(n)$ edges, contains a spanning $(k-1)$-dimensional sphere. This generalises Dirac's theorem, and confirms a conjecture of Georgakopoulos, Haslegrave, Montgomery, and Narayanan. Unlike typical results in the area, our proof does not rely on the absorption method or the regularity lemma. Instead, we use a recently introduced framework that is based on covering the vertex set of the host hypergraph with a family of complete blow-ups.

This is joint work with Freddie Illingworth, Richard Lang, Olaf Parczyk, and Amedeo Sgueglia.

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.