# Seminar

I am one of the members organizing Incheon Combinatorics Seminar. We organize meetings in Thursday (Incheon National University, Inha University) or Friday (SUNY Korea) for this spring semester. We are happy to invite you to give a talk. Please e-mail me if you are interested in.

DATE : 5th, Apr, 2018 (4-5pm).
TITLE: Sharp bound on the number of maximal sum-free subsets of integers
SPEAKER : Hong Liu

ABSTRACT : Cameron and Erd\H{o}s asked whether  the number of \emph{maximal} sum-free sets in $\{1, \dots , n\}$ is much smaller than the  number of sum-free sets. In the same paper they gave a lower bound of $2^{\lfloor n/4 \rfloor }$ for the number of maximal sum-free sets. We resolved this problem, proving that for each $1\leq i \leq 4$, there is a constant $C_i$ such that, given any $n\equiv i \mod 4$, $\{1, \dots , n\}$ contains  $(C_i+o(1)) 2^{n/4}$ maximal sum-free sets.

Our proof makes use of  container and removal lemmas of Green,  a structural result of Deshouillers, Freiman, S\’os and Temkin and a recent bound on the number of subsets of integers with small sumset by Green and Morris. We also discuss related results and open problems on the number of maximal sum-free subsets of abelian groups. Joint work with Jozsef Balogh, Maryam Sharifzadeh and Andrew Treglown.

DATE : 19th, Apr, 2018 (4:20-5:10pm).
TITLE: Forbidding induced even cycles in a graph: typical structure and counting
SPEAKER : Jaehoon Kim

ABSTRACT : We determine, for all k6, the typical structure of graphs that do not contain an induced 2k-cycle. This verifies a conjecture of Balogh and Butterfield. Surprisingly, the typical structure of such graphs is richer than that encountered in related results. The approach we take also yields an approximate result on the typical structure of graphs without an induced 8-cycle or without an induced 10-cycle.