Program

■ January 5 (Saturday)

10:00 – 10:50 Talk 1. Guanghui Wang

11:00 – 11:50 Talk 2. Maryam Sharifzadeh

12:00 – 14:00 Lunch

14:00 – 14:50 Talk 3. Peter Pach

15:00 – 15:50 Talk 4. Hong Liu

15:50 – 16:20 Coffee break

16:20 – 17:10 Talk 5. Richard Montgomery

18:00 – 20:00 Dinner

■ Talk information

Talk 1. Properly colored subgraphs in edge-colored graphs. (Guanghui Wang)

Let $G$ be an edge-colored graph. Given a vertex $v\in V(G)$, the color degree $d^c(v)$ is the number of distinct colors of edges incident to $v$. The minimum color degree $\delta^c(G)$ of $G$ is the minimum $d^c(v)$ over all vertices $v$ in $G$. We say that $G$ is properly colored if no two adjacent edges have the same color. In this talk, we will give some color degree conditions for the existence of properly colored subgraphs. We also mention some related problems.

Talk 2. Number of maximal sum-free subsets of integers and abelian groups. (Maryam Sharifzadeh)

A set $S$ is sum-free if it does not contain a triple $x,y,z$ such that $x+y=z$ (note $x,y$ may not necessarily be distinct). We say that $S\subseteq [n]$ is a maximal sum-free subset of $[n]$ if it is sum-free and it is not properly contained in another sum-free subset of $[n]$. Cameron and Erd\H{o}s asked whether the number of maximal sum-free subsets of $[n]$ is much smaller than the number of sum-free sets. They also gave a lower bound of $2^{\lfloor n/4 \rfloor }$ for the number of maximal sum-free sets. Together with J\'ozsef Balogh, Hong Liu, and Andrew Treglown, we give an exact solution to the problem: For each $1\leq i \leq 4$, there is a constant $C_i$ such that, given any $n\equiv i \mod 4$, $[n]$ contains $(C_i+o(1)) 2^{n/4}$ maximal sum-free sets. In this talk, I will outline the ideas and techniques used in the proof. I will also discuss some new results on the number of maximal sum-free sets in abelian groups, which are joint work with Hong Liu.

Talk 3. The polynomial method and the cap set problem. (Peter Pach)

In this talk we will look at a new variant of the polynomial method which was first used to prove that sets avoiding 3-term arithmetic progressions in groups like $\mathbb{Z}_4^n$ (Croot, Lev and myself) and $\mathbb{Z}_3^n$ (Ellenberg and Gijswijt) are exponentially small (compared to the size of the group).

We will discuss lower and upper bounds for the size of the extremal subsets, including some recent bounds found by Elsholtz and myself. We will also mention some further applications of the method, for instance, the solution of the Erd\H{o}s-Szemer\'edi sunflower conjecture.

Talk 4. Polynomial Schur’s Theorem. (Hong Liu)

I will discuss the Ramsey problem for $\{x,y,z: x+y=p(z)\}$ for polynomials $p$ over $\mathbb(Z)$.

Joint work with Peter Pach and Csaba Sandor.

Talk 5. Latin squares and rainbow subgraphs. (Richard Montgomery)

Latin squares are a combinatorial object with a long history, dating back at least to their study by Euler in the 18th century. A Latin square is an n by n grid filled with n different symbols, so that no symbol appears twice in any row or column. Thus, the underlying grid of any Sudoku puzzle is a 9 by 9 Latin square.

Latin squares can be studied by working on rainbow subgraphs of properly coloured graphs. A graph is properly coloured if its edges are coloured with no two touching edges having the same colour; a rainbow subgraph has a different colour on each of its edges.

I will discuss some recent results on Latin squares and on rainbow subgraphs.

This is joint work with Alexey Pokrovskiy and Benny Sudakov.