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# Paul Seymour, The Erdős-Hajnal conjecture is true for excluding a five-cycle

## Wednesday, December 30, 2020 @ 10:00 AM - 11:00 AM KST

Zoom ID: 869 4632 6610 (ibsdimag)

### Speaker

Paul Seymour
Department of Mathematics, Princeton University
https://www.math.princeton.edu/~pds/

In an n-vertex graph, there must be a clique or stable set of size at least $C\log n$, and there are graphs where this bound is attained. But if we look at graphs not containing a fixed graph H as an induced subgraph, the largest clique or stable set is bigger.

Erdős and Hajnal conjectured in 1977 that for every graph H, there exists c>0 such that every H-free graph has a clique or stable set of size at least $|G|^c$ (“H-free” means not containing H as an induced subgraph, and |G| means the number of vertices of G). This is still open, even for some five-vertex graphs H; and the case that has attracted most attention is when H is a cycle of length five.

It is true in that case. We will give a sketch of the proof, which is via applying a lemma about bipartite graphs, a variant of a theorem of I. Tomon.

This lemma has several other applications to the Erdős-Hajnal conjecture. For instance, it implies that for every cycle C and forest T, there exists c>0 such that every graph that is both C-free and T’-free (where T’ is the complement of T) has a clique or stable set of size $|G|^c$. (Until now this was open when C has length five and T is a 5-vertex path.)

Joint work with Maria Chudnovsky, Alex Scott and Sophie Spirkl.

## Details

Date:
Wednesday, December 30, 2020
Time:
10:00 AM - 11:00 AM KST
Event Category:
Event Tags:

## Venue

Zoom ID: 869 4632 6610 (ibsdimag)

## Organizer

O-joung Kwon (권오정)
기초과학연구원 수리및계산과학연구단 이산수학그룹
대전 유성구 엑스포로 55 (우) 34126
IBS Discrete Mathematics Group (DIMAG)
Institute for Basic Science (IBS)
55 Expo-ro Yuseong-gu Daejeon 34126 South Korea
E-mail: dimag@ibs.re.kr, Fax: +82-42-878-9209