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Some remarks on Folkman graphs for triangles

Eion Mulrenin

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 Erdős 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.

Some remarks on Folkman graphs for triangles

Abstract

Folkman's theorem asserts the existence of graphs which are -free, but which have the property that every two-coloring of contains a monochromatic triangle. The quantitative aspects of , the least such that there exists an -vertex graph with both properties above, are notoriously difficult; a series of improvements over the span of two decades witnessed the solution to two \f(2,3,4) \leq 786H_qH_3q \geq 4\mathscr{T}_qH_qK_4H_qE(H_q)\mathscr{T}_qH_3H_qK_4q$, maintain the Ramsey property with high probability.

Paper Structure

This paper contains 11 sections, 14 theorems, 31 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on the Hermitian unital
  3. The intersection graph $H_q$
  4. The random block construction and non-degenerate triangles
  5. Cliques in $H_q$
  6. Notation
  7. Proof of Theorem \ref{['thm: main-1']}
  8. Proof of Theorem \ref{['thm: main-2']}
  9. The quantitative aspects of Theorem \ref{['thm: main-2']}
  10. Concluding remarks
  11. Acknowledgments

Key Result

Theorem 1.1

For all prime powers $q \geq 4$, there exists graph $H_q$ on $q^4-q^3+q^2$ vertices with a system $\mathscr{T}_q \subseteq \binom{H_q}{K_3}$ of triangles such that Moreover, as $q \to \infty$, every two-coloring of $E(H_q)$ induces at least $(\frac{1}{4}-o(1))|\mathscr{T}_q|$ monochromatic triangles in $\mathscr{T}_q$.

Figures (2)

  • Figure 1: The four lines and six points form an O'Nan configuration.
  • Figure 2:

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • ...and 12 more