## “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

## 2020 Combinatorics Workshop (2020 조합론 학술대회) was held on August 24 online

On August 24, Monday, the 2020 Combinatorics Workshop (2020 조합론 학술대회) was held online due to the COVID-19 pandemic. This local workshop series began in 2004 and has been continued to be one of the biggest annual gathering of people in combinatorics located in Korea. Due to the COVID-19 pandemic, it has been reduced to a one-day online conference on Zoom. It was hosted by Kyung Hee University and IBS Discrete Mathematics Group.

The workshop website: https://cw2020.combinatorics.kr

## There were 5 invited speakers.

• Sejeong Bang (방세정), Yeungnam University, Geometric distance-regular graphs
• Ringi Kim (김린기), KAIST, Decomposing planar graphs into graphs with degree restrictions
• Sangwook Kim (김상욱), Chonnam National University, Combinatorics of lattice path matroid polytopes
• Jinyoung Park (박진영), Institute for Advanced Study, Tuza’s Conjecture for random graphs
• Jongyook Park (박종육), Kyungpook National University, On distance-regular graphs with induced subgraphs $K_{r,t}$

## There were 4 contributed talks.

• Byung-Hak Hwang (황병학), Seoul National University, Acyclic orientation polynomials
• Jaeseong Oh (오재성), Seoul National University, On linearization coefficients of q-Laguerre polynomials
• Jun Seok Oh (오준석), Incheon National University, An inverse Erdős-Ginzburg-Ziv theorem for finite groups
• Tuan Tran, IBS Discrete Mathematics Group, The singularity of random combinatorial matrices

## 2020 Combinatorics Workshop

Combinatorics Workshop (조합론 학술대회) is the biggest annual conference in combinatorics in Korea. It was firstly held in 2004 by the Yonsei University BK21 Research Group. It has been advised by the committee of discrete mathematics of the Korean Mathematical Society since 2013. The aim of this workshop is to bring active researchers with different backgrounds to discuss recent and prospective advances in combinatorics and related areas.

Originally, we planned an offline workshop. However, COVID 19 is more spreading and many participants are worried about attending an offline conference. So the schedule and venue are changed as an online workshop with Zoom. I hope that all participants generously understand this sudden change.

# Contributed speakers

• Byung-Hak Hwang (황병학), Seoul National University
• Jaeseong Oh (오재성), Seoul National University
• Jun Seok Oh (오준석), Incheon National University
• Tuan Tran, Institute for Basic Science (IBS)

## Schedule

### Feb. 11, 2019, Monday

1:30pm-2:20pm Jeong Han Kim: Entropy and sorting
2:20pm-3:10pm Cory T. PalmerGeneralized Turán problems – Berge hypergraphs
Coffee Break
4:00pm-4:50pm Martin BalkoRamsey numbers of edge-ordered graphs
4:50pm-5:40pm Dong Yeap Kang: On the rational Turán exponents conjecture
Banquet

### Feb. 12, 2019, Tuesday

9:30am-10:20am Boram ParkSum-free set problem on integers
Coffee Break
11:00am-11:50am Dániel GerbnerGeneralized Turán problems – counting subgraphs
Lunch

We plan to provide meals to all participants and provide a room at a near-by hotel for invited speakers and selected participants. Please register in the following form below by January 28, Monday; please register early if you want to receive the support for the accommodation.

## Abstracts

#### Jeong Han Kim (김정한), Entropy and Sorting

We reconsider the old problem of sorting under partial information, and give polynomial time algorithms for the following tasks: (1) Given a partial order P, find (adaptively) a sequence of comparisons (questions of the form, “is x < y?”) which sorts ( i.e., finds an unknown linear extension of) P using O(log(e(P))) comparisons in worst case (where e(P) is the number of linear extensions of P). (2) Compute (on line) answers to any comparison algorithm for sorting a partial order P which force the algorithm to use Ω(log(e(P))) comparisons. (3) Given a partial order P of size n, estimate e(P) to within a factor exponential in n. (We give upper and lower bounds which differ by the factor $n^n /n!$.) Our approach, based on entropy of the comparability graph of P and convex minimization via the ellipsoid method, is completely different from earlier attempts to deal with these questions.

Joint work with J. Kahn.

#### Cory T. Palmer, Generalized Turán problems – Berge hypergraphs

Let $F$ be a graph. We say that a hypergraph $H$ is a Berge-$F$ if there is a bijection $f : E(F) \rightarrow E(H )$ such that $e \subseteq f(e)$ for every $e \in E(F)$. Note that Berge-$F$ actually denotes a class of hypergraphs. The maximum number of edges in an $n$-vertex $r$-graph with no subhypergraph isomorphic to any Berge-$F$ is denoted $\operatorname{ex}_r(n,\textrm{Berge-}F)$. Observe that when $r=2$, then a Berge-$F$ is simply the graph $F$ and thus again we! are investigating the Tur\’an function $\operatorname{ex}(n,F)$.

In this talk we will survey results on the function $\operatorname{ex}_r(n,\textrm{Berge-}F)$ for various graphs $F$. We will also describe several interesting open problems.

#### Martin Balko, Ramsey numbers of edge-ordered graphs

An edge-ordered graph is a graph with linearly ordered set of edges. We introduce and study Ramsey numbers of edge-ordered graphs, called edge-ordered Ramsey numbers. We prove some basic properties of these numbers for general edge! -ordered graphs and we provide some stronger estimates for special classes of edge-ordered graphs. We also pose some new open problems and compare edge-ordered Ramsey numbers with the standard Ramsey numbers of graphs and with ordered Ramsey numbers, which are Ramsey numbers for graphs with linearly ordered vertex sets.

Joint work with Mate Vizer.

#### Dong Yeap Kang (강동엽), On the rational Turán exponents conjecture

The extremal number ${\rm ex}(n,F)$ of a graph $F$ is the maximum number of edges in an $n$-vertex graph not containing $F$ as a subgraph. A real number $r \in [1,2]$ is realisable if there exists a graph $F$ with ${\rm ex}(n , F) = \Theta(n^r)$. Several decades ago, Erdős and Simonovits conjectured that every rational number in $[1,2]$ is realisable. Despite decades of effort, the only known realisable numbers are $1,\frac{7}{5},2$, and the numbers of the form $1+\frac{1}{m}$, $2-\frac{1}{m}$, $2-\frac{2}{m}$ for integers $m \geq 1$. In particular, it is not even known whether the set of all realisable numbers contains a single limit point other than two numbers 1 and 2.

We discuss some recent progress on the conjecture of Erdős and Simonovits. First, we show that $2 – \frac{a}{b}$ is realisable for any integers $a,b \geq 1$ with $b>a$ and $b \equiv \pm 1 ~({\rm mod}\:a)$. This includes all previously known ones, and gives infinitely many limit points $2-\frac{1}{m}$ in the set of all realisable numbers as a consequence.

Secondly, we propose a conjecture on subdivisions of bipartite graphs. Apart from being interesting on its own, we show that, somewhat surprisingly, this subdivision conjecture in fact implies that every rational number between 1 and 2 is realisable.

This is joint work with Jaehoon Kim and Hong Liu.

#### Boram Park (박보람), Sum-free set problem on integers

For an abelian  group $G$, a set $A \subset G$ is sum-free if there are no $x, y, z \in A$ such that $x + y = z$. Sum-free sets was initiated by Schur (1916) by an attempt to prove the famous Fermat’s Last Theorem. Since then, there have been intensive and fruitful research in the field of additive combinatorics. One of great interest in the study of sum-free sets is to consider sum-free subsets of a set of integers, which has attracted a significant attention in the literature over the years.

In this talk, some recent results on sum-free sets of integers are discussed. Then we present a result on $k$-sum $\bf{n}$-free set, where $\bf{n}$ is an $n$-dimensional integer vector. The work is based on joint work with Ilkyoo Choi and Ringi Kim.

#### Dániel Gerbner, Generalized Turán problems – counting subgraphs

Given two graphs $H$ and $F$, our goal is to determine the maximum number of copies of $H$ in an $F$-free graph on $n$ vertices. The systematic research of these problems was initiated (after several sporadic results) by Alon and Shikhelman. I describe several results of mine in this area, with different sets of co-authors.

Joint work with Ervin Győri, Abhishek Methuku, Cory Palmer and Mate Vizer.

## 2019-1 IBS Workshop on Graph Theory (Feb. 11-12, 2019). Registration due: Jan. 28

The Discrete Mathematics Group (DIMAG) at the Institute for Basic Science (IBS) will have its first workshop in Feb. 11-12, 2019. If you want to participate, then please register by January 28 on the website. We plan to provide accommodation for speakers and selected participants requested during the online registration.