School of Mathematical Sciences Academic Report [2025] No. 036  

(High-Level University Construction Series Report No. 1058)  

Lecture Title: Density Hajnal—Szemerédi theorem for cliques of size four

Speaker: Hou Jianfeng (Professor, Fuzhou University)  

Date & Time: 10:00–11:00 AM, May 17, 2025

Venue: Room 501, Huixing Building  

Abstract: The celebrated Corrádi—Hajnal Theorem and the Hajnal—Szemerédi Theorem determined the exact minimum degree thresholds for a graph on n vertices to contain k vertex-disjoint copies of K_{r}, for r=3 and general r≥4, respectively. The edge density version of the Corrádi—Hajnal Theorem was established by Allen—Böttcher—Hladký—Piguet for large n. Remarkably, they determined the four classes of extremal constructions corresponding to different intervals of k. They further proposed the natural problem of establishing a density version of the Hajnal— Szemerédi Theorem: For r≥4, what is the edge density threshold that guarantees a graph on n vertices contains k vertex-disjoint copies of K_{r} for k≤n/r. They also remarked, "We are not even sure what the complete family of extremal graphs should be."

We take the first step toward this problem by determining asymptotically the five classes of extremal constructions for r=4. Furthermore, we propose a candidate set comprising r+1 classes of extremal constructions for general r≥5.

Biography: Hou Jianfeng is a professor and Ph.D. supervisor at Fuzhou University. He received his Ph.D. in Mathematics from Shandong University in July 2009. He was nominated for the National Excellent Doctoral Dissertation Award in 2011 and awarded the Distinguished Young Scholars Project of the Fujian Provincial Natural Science Foundation in the same year. In 2020, he was selected as a young talent in Fujian Province's "Eagle Plan." He has presided over 1 national key R&D project, 4 National Natural Science Foundation projects, and participated in 1 key project. His main research focuses on graph theory and its applications, with over 60 published papers. He serves as the Secretary-General of the Combinatorics and Graph Theory Professional Committee of the Chinese Mathematical Society and an Executive Member of the Graph Theory, Combinatorics, and Applications Professional Committee of the Chinese Society for Industrial and Applied Mathematics.  

All faculty and students are welcome to attend!

Invited by: Huang Zejun

School of Mathematical Sciences  

May 12, 2025