I am interested in various topics related to extremal and probabilistic combinatorics. Some of these are Ramsey theory, random graphs, (hyper-)graph partition problems, positional games and combinatorial geometry.
We study the positional game where two players, Maker and Breaker, alternately select respectively $1$ and $b$ previously unclaimed edges of $K_n$. Maker wins if she succeeds in claiming all edges of some odd cycle in $K_n$ and Breaker wins otherwise. Improving on a result of Bednarska and Pikhurko, we show that Maker wins the odd cycle game if $b \leq \big( (4 - \sqrt6)/5 + o(1) \big) \, n$. We furthermore introduce ``connected rules'' and study the odd cycle game under them, both in the Maker-Breaker as well as in the Client-Waiter variant.
Extending a result of Rado to hypergraphs, we prove that for all $r, k, t \in \mathbb{N}$ with $k \geq t \geq 2$, the vertices of every $r(k-t+1)$-edge-coloured countably infinite complete $k$-graph can be core-partitioned into at most $r$ monochromatic $t$-tight Berge-paths of different colours. We further describe a construction showing that this result is best possible.
Confirming a conjecture of Gyárfás, we prove that, for all natural numbers $k$ and $r$, the vertices of every $r$-edge-coloured complete $k$-uniform hypergraph can be partitioned into a bounded number (independent of the size of the hypergraph) of monochromatic tight cycles. We further prove that, for for all natural numbers $p$ and $r$, the vertices of every $r$-edge-coloured complete graph can be partitioned into a bounded number of $p$-th powers of cycles, settling a problem of Elekes, Soukup, Soukup and Szentmiklóssy. In fact we prove a common generalisation of both theorems which further extends these results to all host hypergraphs of bounded independence number.
We prove that in every $2$-colouring of the edges of $K_{\mathbb N}$ there exists a monochromatic (one-way infinite) path $P$ such that $V(P)$ has upper density at least $(12+\sqrt{8})/17 \approx 0.87226$ and further show that this is best possible. This settles a problem of Erdős and Galvin.
A finite set $A \subset \mathbb{R}^d$ is called diameter-Ramsey if for every $r \in \mathbb N$, there exists some $n \in \mathbb N$ and a finite set $B \subset \mathbb{R}^n$ with $\mathrm{diam}(A)=\mathrm{diam}(B)$ such that whenever $B$ is coloured with $r$ colours, there is a monochromatic set $A' \subset B$ which is congruent to $A$. We prove that sets of diameter $1$ with circumradius larger than $1/\sqrt{2}$ are not diameter-Ramsey. In particular, we obtain that triangles with an angle larger than $135^\circ$ are not diameter-Ramsey, improving a result of Frankl, Pach, Reiher and Rödl. Furthermore, we deduce that there are simplices which are almost regular but not diameter-Ramsey.
We prove balanced supersaturation theorems for $\theta_{a,b}$ (the graph consisting of $a$ internally vertex-disjoint paths of length $b$, each with the same endpoints) and the complete $r$-partite $r$-uniform hypergraph $K_{a_1,\ldots,a_r}^{(r)}$. We use the hypergraph container method to deduce that, for every $a,b \geq 2$, there are at most $2^{O(n^{1+1/b})}$ $\theta_{a,b}$-free graphs on $n$ vertices, and that at most $2^{o(n^{1+1/b})}$ of them have $o(n^{1+1/b})$ edges. Similarly we deduce that, for all $2 \leq a_1 \leq \ldots \leq a_r$, there are at most $2^{O\left(n^{r-1/(a_1 \cdots a_{r-1})}\right)}$ $K_{a_1, \ldots, a_r}^{(r)}$-free $r$-graphs on $n$ vertices, and that at most $2^{o\left(n^{r-1/(a_1 \cdots a_{r-1})}\right)}$ of them have $o\left(n^{r-1/(a_1 \cdots a_{r-1})}\right)$ edges. For most $\theta$-graphs and for most complete $r$-partite $r$-graphs, these results are optimal up to the implicit constant factor in the exponent.
The grid Ramsey number $ G(r) $ is the smallest number $ n $ such that every edge-colouring of the grid graph $\Gamma_{n,n} := K_n \times K_n$ with $r$ colours induces a rectangle whose parallel edges receive the same colour. We show $ G(r) \leq r^{\binom{r+1}{2}} - \left( 1/4 - o(1) \right) r^{\binom{r}{2}+1} $, slightly improving the currently best known upper bound due to Gyárfás.
Sebastián Bustamante, Louis DeBiasio, Nóra Frankl, Ander Lamaison, Richard Lang, Adva Mond, Alexey Pokrovskiy, Jozef Skokan, Christoph Spiegel, Tibor Szabó, Tuan Tran