Some remarks on Folkman graphs for triangles

Abstract: Folkman's theorem asserts the existence of graphs $G$ which are $K_4$-free, but which have the property that every two-coloring of $E(G)$ contains a monochromatic triangle. The quantitative aspects of $f(2,3,4)$, the least $n$ such that there exists an $n$-vertex graph with both properties above, are notoriously difficult; a series of improvements over the span of two decades witnessed the solution to two \$100 Erd\H{o}s problems, and the current record due to Lange, Radziszowski, and Xu now stands at $f(2,3,4) \leq 786$, the proof of which is computer-assisted. In this paper, we study Folkman-like properties of a sequence $H_q$ of finite geometric graphs constructed using Hermitian unitals in projective planes which were instrumental in the recent Mattheus-Verstra\"ete breakthrough on off-diagonal Ramsey numbers. We show that for all prime powers $q \geq 4$, there exists a subset $\mathscr{T}_q$ of triangles in $H_q$ such that no four span a $K_4$ in $H_q$, but every two-coloring of $E(H_q)$ induces a monochromatic triangle in $\mathscr{T}_q$. For $q=4$, this gives a graph on $208$ vertices with this "quasi-Folkman" property. Moreover, we show that a certain random alteration of $H_q$ which destroys all of its $K_4$'s will, for large $q$, maintain the Ramsey property with high probability.