Work

Optimally pseudorandom $K_4$-free graphs

Year: 2026

Type: preprint

Abstract: We show that optimally pseudorandom $K_4$-free graphs of order $n$ and degree $d = Θ(n^{4/5})$ exist by constructing a graph in the split Cayley hexagon, matching the known upper bound. This resolves the first open case for $K_k$-free graphs after $k=3$ for which Alon gave a tight construction in 1994. This has a variety of implications for $K_4$-free pseudorandom graphs. Furthermore, it implies an explicit lower bound on the Ramsey number of $r(4, t) \geq t^{1.\overline{6} - o(1)}$, improving the previous record by Kostochka, Pudlák, and Rödl of $r(4, t) \geq t^{1.6-o(1)}$.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2609.32461

Authors: Han, Jie, Ihringer, Ferdinand, Van Maldeghem, Hendrik

Institutions: Beijing Institute of Technology, Southern University of Science and Technology, Ghent University

Language: English

Topic: Limits and Structures in Graph Theory

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Natural Science Foundation of China, Fundamental Research Funds for the Central Universities

Award: G3977112998

OpenAlex ID: W7214733656

Cites: 2

Cited by: 2

Related to: 10