Some remarks on Folkman graphs for triangles

TL;DR

This paper explores Folkman graphs related to geometric constructions, conjectures the existence of a Folkman subgraph within a specific graph, and demonstrates that certain random modifications preserve key properties.

Contribution

It introduces a sequence of geometric graphs with Folkman-like properties and provides evidence for a conjecture about a specific 63-vertex graph containing a Folkman graph.

Findings

Existence of triangle systems in $H_q$ with no $K_4$ spanning four vertices.

Verification of properties for $H_3$ through computational methods.

Random alterations of $H_q$ maintain Ramsey properties with high probability for large $q$.

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, and conjecture that the graph $H_3$, which has 63 vertices, contains a Folkman graph as a proper subgraph. As evidence towards this conjecture, we show that for all prime powers $q \geq 4$, there exists a system $\mathscr{T}_q$ 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 $\mathscr{T}_q$, and note that recent adjacent computational work has verified the same property for $H_3$. 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.