New bounds on the vertex Folkman number $F_v(2, 2, 2, 3; 4)$

TL;DR

This paper establishes tighter bounds on the vertex Folkman number $F_v(2, 2, 2, 3; 4)$, demonstrating it lies between 20 and 22 through computational methods, advancing understanding of graph coloring and clique avoidance.

Contribution

The authors improve the bounds on $F_v(2, 2, 2, 3; 4)$ by computationally proving it is between 20 and 22, refining previous estimates.

Findings

Established that $20 \,\leq\, F_v(2, 2, 2, 3; 4) \leq 22$

Used computer-assisted proofs to refine bounds

Contributed to the understanding of vertex Folkman numbers

Abstract

For a graph $G$ the expression $G \overset{v}{\rightarrow} (a_1, ..., a_s)$ means that for every coloring of the vertices of $G$ in $s$ colors there exists $i \in \{1, ..., s\}$ such that there is a monochromatic $a_i$-clique of color $i$. The vertex Folkman number $F_v(a_1, ..., a_s; q)$ is defined as $$F_v(a_1, ..., a_s; q) = \min\{\vert V(G) \vert : G \overset{v}{\rightarrow} (a_1, ..., a_s) \mbox{ and } K_q \not\subseteq G\}.$$ In this paper we improve the known bounds on the number $F_v(2, 2, 2, 3; 4)$ by proving with the help of a computer that $20 \leq F_v(2, 2, 2, 3; 4) \leq 22$.