# Jan Corsten [CV]

Since January 2017, I am a PhD student at the Maths Department of The London School of Economics and Political Science, where I am supervised by Peter Allen and Julia Böttcher. I obtained my B.Sc. and M.Sc. at Freie Universität Berlin, where I was supervised by Shagnik Das and Tibor Szabó in my final year.
You can contact me at j.corsten@lse.ac.uk.

### Research Interests

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.

### Papers

1. On the Odd Cycle Game and Connected Rules [arXiv] With Adva Mond, Alexey Pokrovskiy, Christoph Spiegel and Tibor Szabó, 2019, submitted.
Abstract

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.

2. Partitioning infinite hypergraphs into few monochromatic Berge-paths [arXiv] With Sebastián Bustamante and Nóra Frankl, 2019, to appear in "Graphs and Combinatorics"
Abstract

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.

3. Partitioning edge-coloured hypergraphs into few monochromatic tight cycles [arXiv] With Sebastián Bustamante, Nóra Frankl, Alexey Pokrovskiy and Jozef Skokan, 2019, submitted.
Abstract

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.

4. Upper density of monochromatic infinite paths [arXiv] With Louis DeBiasio, Ander Lamaison and Richard Lang, “Advances in Combinatorics (2019), doi
Abstract

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.

5. A note on diameter-Ramsey sets [arXiv] With Nóra Frankl, “European Journal of Combinatorics 61 (2018), 51-54, doi”.
Abstract

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.

6. Balanced supersaturation for degenerate hypergraphs [arXiv] With Tuan Tran, 2017, submitted.
Abstract

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.

7. A note on the grid Ramsey problem [arxiv] “Electronic Notes in Discrete Mathematics 61 (2017), 287-292, doi”.
Abstract

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.

### Co-authors

Sebastián Bustamante, Louis DeBiasio, Nóra Frankl, Ander Lamaison, Richard Lang, Adva Mond, Alexey Pokrovskiy, Jozef Skokan, Christoph Spiegel, Tibor Szabó, Tuan Tran