I am interested in various topics related to extremal combinatorics. Some of these are Ramsey theory, (hyper-)graphs with forbidden subgraphs, partitioning (hyper-)graphs into large subgraphs, positional games and combinatorial geometry.
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.
Louis DeBiasio, Ander Lamaison, Richard Lang, Nóra Frankl, Tuan Tran