A notion of sublinear expander has played a central role in the resolutions of a couple of long-standing conjectures in embedding problems in graph theory, including e.g. the odd cycle problem of Erdős and Hajnal that the harmonic sum of odd cycle length in a graph diverges with its chromatic number. I will survey some of these developments.

## Extremal and Probabilistic Combinatorics (2021 KMS Spring Meeting)

A special session “Extremal and Probabilistic Combinatorics” at the 2021 KMS Spring Meeting is organized by Tuan Tran.

URL: https://www.kms.or.kr/meetings/spring2021/

## Speakers and Schedule

All talks are on April 30.

- [9:00 am]
**Joonkyung Lee (이준경)**, University College London*Majority dynamics on sparse random graphs*

- [9:30 am]
**Dong Yeap Kang (강동엽)**, Unversity of Birmingham*The Erdős-Faber-Lovász conjecture and related results*

- [10:00 am]
**Jinyoung Park (박진영)**, IAS*The threshold for the square of a Hamilton cycle*

- [10:50 am]
**Debsoumya Chakraborti**, IBS Discrete Mathematics Group*Generalized graph saturation*

- [11:20 am]
**Jaehoon Kim (김재훈)**, KAIST*Resolution of the Oberwolfach problem*

- [11:50 am]
**Hong Liu**, University of Warwick*Sublinear expanders and its applications*

## Abstracts

#### Debsoumya Chakraborti, Generalized graph saturation

Graph saturation is one of the oldest areas of investigation in extremal combinatorics. A graph G is called *F-saturated* if G does not contain a subgraph isomorphic to F, but the addition of any edge creates a copy of F. We resolve one of the most fundamental questions of minimizing the number of cliques of size r in a $K_s$-saturated graph for all sufficiently large numbers of vertices, confirming a conjecture of Kritschgau, Methuku, Tait and Timmons. We further prove a corresponding stability result. This talk will be based on joint work with Po-Shen Loh.

#### Jaehoon Kim (김재훈), Resolution of the Oberwolfach problem

The Oberwolfach problem, posed by Ringel in 1967, asks for a decomposition of $K_{2n+1}$ into edge-disjoint copies of a given 2-factor. We show that this can be achieved for all large n. We actually prove a significantly more general result, which allows for decompositions into more general types of factors.

#### Dong Yeap Kang (강동엽), The Erdős-Faber-Lovász conjecture and related results

A hypergraph is linear if every pair of two distinct edges shares at most one vertex. A longstanding conjecture by Erdős, Faber, and Lovász in 1972, states that the chromatic index of any linear hypergraph on n vertices is at most n.

In this talk, I will present the ideas to prove the conjecture for all large n. This is joint work with Tom Kelly, Daniela Kühn, Abhishek Methuku, and Deryk Osthus.

#### Joonkyung Lee (이준경), Majority dynamics on sparse random graphs

*Majority dynamics* on a graph G is a deterministic process such that every vertex updates its {-1,1}-assignment according to the majority assignment on its neighbor simultaneously at each step. Benjamini, Chan, O’Donnell, Tamuz and Tan conjectured that, in the Erdős-Rényi random graph G(n,p), the random initial {-1,1}-assignment converges to the unanimity with high probability whenever p>> 1/n.

This conjecture was firstly confirmed for $p>Cn^{-1/2}$ for a large constant C>0 by Fountoulakis, Kang and Makai. Although this result has been reproved recently by Tran and Vu and by Berkowitz and Devlin, none of them managed to extend it beyond the barrier $p>Cn^{-1/2}$. We prove the conjecture for sparser random graphs G(n,p), where $Dn^{-3/5}\log n < p < C n^{-1/2}$ with a large constant D>0.

Joint work with Debsoumya Chakraborti, Jeong Han Kim and Tuan Tran.

#### Hong Liu, Sublinear expanders and its applications

I will review the history of sublinear expander and present some recent applications, which lead to resolutions of several long-standing problems in sparse graphs embeddings.

#### Jinyoung Park (박진영), The threshold for the square of a Hamilton cycle

We will talk about a recent result of Jeff Kahn, Bhargav Narayanan, and myself stating that the threshold for the random graph G(n,p) to contain the square of a Hamilton cycle is $1/\sqrt n$, resolving a conjecture of Kühn and Osthus from 2012. The proof idea is motivated by the recent work of Frankston and the three aforementioned authors on a conjecture of Talagrand — “a fractional version of Kahn-Kalai expectation threshold conjecture.”

## Hong Liu gave a talk on the extremal problem of finding nested cycles with no geometric crossing at the Discrete Math Seminar

On March 22, 2021 at the Discrete Math Seminar, Hong Liu (刘鸿) from the University of Warwick gave a talk on the extremal problem of having nested cycles with no geometric crossing by using the sublinear expanders. The title of his talk was “Nested cycles with no geometric crossing“.

## Hong Liu (刘鸿), Nested cycles with no geometric crossing

In 1975, Erdős asked the following question: what is the smallest function $f(n)$ for which all graphs with $n$ vertices and $f(n)$ edges contain two edge-disjoint cycles $C_1$ and $C_2$, such that the vertex set of $C_2$ is a subset of the vertex set of $C_1$ and their cyclic orderings of the vertices respect each other? We prove the optimal linear bound $f(n)=O(n)$ using sublinear expanders.

This is joint work with Irene Gil Fernández, Jaehoon Kim and Younjin Kim.

## Hong Liu presented his result with R. Montgomery on the resolution the odd cycle problem of Erdős and Hajnal at the Discrete Math Seminar

On December 8, 2020, Hong Liu from University of Warwick presented a talk on his recent work with Richard Montgomery, answering the odd cycle problem of Erdős and Hajnal (1981). The title of his talk is “A solution to Erdős and Hajnal’s odd cycle problem“.

## Hong Liu (刘鸿), A solution to Erdős and Hajnal’s odd cycle problem

I will go over the history on the study of the set of cycle lengths of graphs with large average degree or chromatic number, and discuss recent work with Richard Montgomery on this topic. In particular, we will see the divergence of harmonic sum of odd cycle lengths in graphs with large chromatic number and the appearance of cycle lengths in very sparse sequences (such as powers of 2). The methods developed in this work allows also us to embed equally divided clique subdivisions, which was conjectured by Thomassen.

## Hong Liu (刘鸿), Cycles and trees in graphs (8/8)

This lecture series covers several different techniques on embedding paths/trees/cycles in (pseudo)random graphs/expanders as (induced) subgraphs.

## Hong Liu (刘鸿), Cycles and trees in graphs (7/8)

This lecture series covers several different techniques on embedding paths/trees/cycles in (pseudo)random graphs/expanders as (induced) subgraphs.

## Hong Liu (刘鸿), Cycles and trees in graphs (6/8)

This lecture series covers several different techniques on embedding paths/trees/cycles in (pseudo)random graphs/expanders as (induced) subgraphs.