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.

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

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.