Optimally pseudorandom $K_4$-free graphs

Abstract

We show that optimally pseudorandom $K_4$-free graphs of order $n$ and degree $d=\Theta (n^{4/5})$ 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 $r(4,t)\ge t^{1.\overline{6}-o(1)}$, improving the previous record by Kostochka, Pudlák, and Rödl of $r(4,t)\ge t^{1.6-o(1)}$.

8 pages, 4 figures, ChatGPT Astra 6 found the construction for q=2^h, h odd