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 $p>C{n}^{-1/2}$ 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 $p>C{n}^{-1/2}$. We prove the conjecture for sparser random graphs G(n,p), where $D{n}^{-3/5}\log n<p<C{n}^{-1/2}$ 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.”