## “2021 Combinatorics Workshop” was held from December 20 to December 22, 2021 at Yangpyeong

The 2021 Combinatorics Workshop (2021 조합론 학술대회) was held from December 20, 2021 to December 22, 2021 at the Bloomvista, Yangpyeong. There were 5 invited talks and 12 contributed talks.

### Speakers of the contributed talks

• Jungho Ahn, KAIST / IBS DIMAG
• Jin-Hwan Cho, NIMS
• Linda Cook, IBS DIMAG
• Cheolwon Heo, Sungkyunkwan University
• Seonghyuk Im, KAIST
• Hyobin Kim, Kyungpook National University
• Minki Kim, IBS DIMAG
• Hyemin Kwon, Ajou University
• Hyunwoo Lee, KAIST
• Sang June Lee, Kyung Hee University
• Jaehyeon Seo, KAIST
• Semin Yoo, KIAS

### Participants (50 people, all of whom are fully vaccinated against COVID-19)

• Jungho Ahn, speaker, KAIST / IBS DIMAG
• Sejeong Bang, session chair, Yeungnam University
• Rutger Campbell, IBS DIMAG
• Debsoumya Chakraborti, IBS DIMAG
• Eun-Kyung Cho, Hankuk University of Foreign Studies
• Hyunsoo Cho, Ewha Womans University
• Jin-Hwan Cho, speaker, NIMS
• Jeong-Ok Choi, organizer/session chair, GIST
• Linda Cook, speaker, IBS DIMAG
• Taehyun Eom, KAIST
• Cheolwon Heo, speaker, Sungkyunkwan University
• Seonghyuk Im, speaker, KAIST
• Jihyeug Jang, Sungkyunkwan University
• Dosang Joe, NIMS
• Donggyu Kim, KAIST / IBS DIMAG
• Donghyun Kim, Sungkyunkwan University
• Dongsu Kim, invited speaker, KAIST
• Hyobin Kim, speaker, Kyungpook National University
• Jaehoon Kim, KAIST
• Jang Soo Kim, Sungkyunkwan University
• Jinha Kim, IBS DIMAG
• Minki Kim, speaker, IBS DIMAG
• Seog-Jin Kim, session chair, Konkuk University
• Doowon Koh, Chungbuk National University
• Hyemin Kwon, speaker, Ajou University
• O-joung Kwon, Incheon National University / IBS DIMAG
• Dabeen Lee, IBS DIMAG
• Duksang Lee, KAIST / IBS DIMAG
• Hyunwoo Lee, speaker, KAIST
• Joonkyung Lee, invited speaker, Hanyang University
• Sang June Lee, speaker, Kyung Hee University
• Seung Jin Lee, Seoul National University
• Hong Liu, invited speaker, University of Warwick, UK
• Ben Lund, IBS DIMAG
• Suil O, invited speaker, SUNY Korea
• Jaeseong Oh, KIAS
• Sang-il Oum, organizer/session chair, IBS DIMAG / KAIST
• Jae Hyun Park, Kyung Hee University
• Seonjeong Park, invited speaker, Jeonju University
• Jaehyeon Seo, speaker, KAIST
• Seunghyun Seo, session chair, Kangwon National University
• Heesung Shin, organizer/session chair, Inha University
• Mark Siggers, Kyungpook National University
• Jaebum Sohn, Yonsei University
• Minho Song, Sungkyunkwan University
• U-keun Song, Sungkyunkwan University
• Jeong Hyun Sung, Seoul National University
• Tuan Tran, IBS DIMAG
• Sounggun Wee, KAIST / IBS DIMAG
• Semin Yoo, speaker, KIAS

## 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 $\pm 1$-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 $\pm 1$-assignment converges to a $99\%$-agreement with high probability whenever $p=\omega(1/n)$.

This conjecture was first confirmed for $p\geq\lambda n^{-1/2}$ for a large constant $\lambda$ by Fountoulakis, Kang and Makai. Although this result has been reproved recently by Tran and Vu and by Berkowitz and Devlin, it was unknown whether the conjecture holds for $p< \lambda n^{-1/2}$. We break this $\Omega(n^{-1/2})$-barrier by proving the conjecture for sparser random graphs $G(n,p)$, where $\lambda’ n^{-3/5}\log n \leq p \leq \lambda n^{-1/2}$ with a large constant $\lambda’>0$.

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

## 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.

## Speakers and Schedule

All talks are on April 30.

## 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.”

## Joonkyung Lee (이준경) gave online talks on the Ramsey multiplicity and common graphs at the Discrete Math Seminar

On November 30 and December 2, 2020, Joonkyung Lee (이준경) from University College London gave two online talks on the Ramsey multiplicity and common graphs at the Discrete Math Seminar organized by Jaehoon Kim at KAIST. The titles of his talks are “On Ramsey multiplicity” and “On common graphs“.

(The photo above was taken earlier in his other seminar talk.)

## Joonkyung Lee (이준경), On common graphs

A graph $H$ is common if the number of monochromatic copies of $H$ in a 2-edge-colouring of the complete graph $K_n$ is minimised by the random colouring. Burr and Rosta, extending a famous conjecture by Erdős, conjectured that every graph is common. The conjectures by Erdős and by Burr and Rosta were disproved by Thomason and by Sidorenko, respectively, in the late 1980s.

Despite its importance, the full classification of common graphs is still a wide open problem and has not seen much progress since the early 1990s. In this lecture, I will present some old and new techniques to prove whether a graph is common or not.

## Joonkyung Lee (이준경), On Ramsey multiplicity

Ramsey’s theorem states that, for a fixed graph $H$, every 2-edge-colouring of $K_n$ contains a monochromatic copy of $H$ whenever $n$ is large enough. Perhaps one of the most natural questions after Ramsey’s theorem is then how many copies of monochromatic $H$ can be guaranteed to exist. To formalise this question, let the Ramsey multiplicity $M(H;n)$ be the minimum number of labelled copies of monochromatic $H$ over all 2-edge-colouring of $K_n$. We define the Ramsey multiplicity constant $C(H)$ is defined by $C(H):=\lim_{n\rightarrow\infty}\frac{M(H,n)}{n(n-1)\cdots(n-v+1)}$. I will discuss various bounds for C(H) that are known so far.

## Joonkyung Lee (이준경) gave a talk on norms defined from graphs at the Discrete Math Seminar

On October 21, 2020, Joonkyung Lee (이준경) from University College London gave a talk at the Discrete Math Seminar on norms defined from graphs motivated by Sidorenko’s conjecture and Gowers norms on extremal combinatorics, unifying two seemingly different concepts of real-norming graphs and complex-norming graphs. The title of his talk was “On graph norms for complex-valued functions“. Joonkyung Lee will stay at the IBS discrete mathematics group for several weeks from October 19.

## Joonkyung Lee (이준경), On some properties of graph norms

For a graph $H$, its homomorphism density in graphs naturally extends to the space of two-variable symmetric functions $W$ in $L^p$, $p\geq e(H)$, denoted by $t_H(W)$. One may then define corresponding functionals $\|W\|_{H}:=|t_H(W)|^{1/e(H)}$ and $\|W\|_{r(H)}:=t_H(|W|)^{1/e(H)}$ and say that $H$ is (semi-)norming if $\|.\|_{H}$ is a (semi-)norm and that $H$ is weakly norming if $\|.\|_{r(H)}$ is a norm.

We obtain some results that contribute to the theory of (weakly) norming graphs. Firstly, we show that ‘twisted’ blow-ups of cycles, which include $K_{5,5}\setminus C_{10}$ and $C_6\square K_2$, are not weakly norming. This answers two questions of Hatami, who asked whether the two graphs are weakly norming. Secondly, we prove that $\|.\|_{r(H)}$ is not uniformly convex nor uniformly smooth, provided that $H$ is weakly norming. This answers another question of Hatami, who estimated the modulus of convexity and smoothness of $\|.\|_{H}$. We also prove that every graph $H$ without isolated vertices is (weakly) norming if and only if each component is an isomorphic copy of a (weakly) norming graph. This strong factorisation result allows us to assume connectivity of $H$ when studying graph norms. Based on joint work with Frederik Garbe, Jan Hladký, and Bjarne Schülke.

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IBS Discrete Mathematics Group (DIMAG)
Institute for Basic Science (IBS)
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E-mail: dimag@ibs.re.kr, Fax: +82-42-878-9209