## Posts Tagged ‘이준경’

### Joonkyung Lee (이준경), Sidorenko’s conjecture for blow-ups

Tuesday, December 25th, 2018

IBS/KAIST Joint Discrete Math Seminar

Sidorenko’s conjecture for blow-ups
Joonkyung Lee (이준경)
Universität Hamburg, Hamburg, Germany
2019/1/3 Thursday 4PM (Room: DIMAG, IBS)
A celebrated conjecture of Sidorenko and Erdős–Simonovits states that, for all bipartite graphs H, quasirandom graphs contain asymptotically the minimum number of copies of H taken over all graphs with the same order and edge density. This conjecture has attracted considerable interest over the last decade and is now known to hold for a broad range of bipartite graphs, with the overall trend saying that a graph satisfies the conjecture if it can be built from simple building blocks such as trees in a certain recursive fashion.Our contribution here, which goes beyond this paradigm, is to show that the conjecture holds for any bipartite graph H with bipartition A∪B where the number of vertices in B of degree k satisfies a certain divisibility condition for each k. As a corollary, we have that for every bipartite graph H with bipartition A∪B, there is a positive integer p such that the blow-up HAp formed by taking p vertex-disjoint copies of H and gluing all copies of A along corresponding vertices satisfies the conjecture. Joint work with David Conlon.

### Joonkyung Lee (이준경), The extremal number of subdivisions

Thursday, September 13th, 2018
The extremal number of subdivisions
Joonkyung Lee (이준경)
Universität Hamburg, Hamburg, Germany
2018/9/17 Monday 5PM
One of the cornerstones of extremal graph theory is a result of Füredi, later reproved and given due prominence by Alon, Krivelevich and Sudakov, saying that if H is a bipartite graph with maximum degree r on one side, then there is a constant C such that every graph with n vertices and C n2 – 1/r edges contains a copy of H. This result is tight up to the constant when H contains a copy of Kr,s with s sufficiently large in terms of r. We conjecture that this is essentially the only situation in which Füredi’s result can be tight and prove this conjecture for r = 2. More precisely, we show that if H is a C4-free bipartite graph with maximum degree 2 on one side, then there are positive constants C and δ such that every graph with n vertices and C n3/2 – δ edges contains a copy of H. This answers a question by Erdős from 1988. The proof relies on a novel variant of the dependent random choice technique which may be of independent interest. This is joint work with David Conlon.

### Joonkyung Lee (이준경), Counting tree-like graphs in locally dense graphs

Monday, January 1st, 2018
Counting tree-like graphs in locally dense graphs
Joonkyung Lee (이준경)
Mathematical Institute, University of Oxford, Oxford, UK
2018/1/8 Mon 4PM-5PM
We prove that a class of graphs obtained by gluing complete multipartite graphs in a tree-like way satisfies a conjecture of Kohayakawa, Nagle, Rödl, and Schacht on random-like counts for small graphs in locally dense graphs. This implies an approximate version of the conjecture for graphs with bounded tree-width. We also prove an analogous result for odd cycles instead of complete multipartite graphs.
The proof uses a general information theoretic method to prove graph homomorphism inequalities for tree-like structured graphs, which may be of independent interest.

### 1st Korean Workshop on Graph Theory

Tuesday, July 28th, 2015
1st Korean Workshop on Graph Theory
August 26-28, 2015
KAIST  (E6-1 1501 & 3435)
http://home.kias.re.kr/MKG/h/KWGT2015/
• Program Book
• Currently, we are planning to have talks in KOREAN.
• Students/postdocs may get the support for the accommodation. (Hotel Interciti)
• Others may contact us if you wish to book a hotel at a pre-negotiated price. Please see the website.
• PLEASE REGISTER UNTIL AUGUST 16.
Location: KAIST
• Room 1501 of E6-1 (August 26, 27)
• Room 3435 of E6-1 (August 28)
Invited Speakers:
Organizers:

### Joonkyung Lee, Some Advances in Sidorenko’s Conjecture

Tuesday, August 26th, 2014
Sidorenko’s conjecture states that for every bipartite graph $$H$$ on $$\{1,\cdots,k\}$$
holds, where $$\mu$$ is the Lebesgue measure on $$[0,1]$$ and $$h$$ is a bounded, non-negative, symmetric, measurable function on $$[0,1]^2$$. An equivalent discrete form of the conjecture is that the number of homomorphisms from a bipartite graph $$H$$ to a graph $$G$$ is asymptotically at least the expected number of homomorphisms from $$H$$ to the Erdos-Renyi random graph with the same expected edge density as $$G$$. In this talk, we will give an overview on known results and new approaches to attack Sidorenko’s conjecture.