Lower bounding the Folkman numbers $F_v(a_1, ..., a_s; m - 1)$

TL;DR

This paper establishes new lower bounds for certain vertex Folkman numbers using an improved algorithm, narrowing the gap between known bounds and providing exact values in specific cases.

Contribution

It introduces a novel method and an improved algorithm to derive lower bounds for Folkman numbers, especially when the maximum parameter is 7.

Findings

Proved that $F_v(a_1, ..., a_s; m - 1) \\geq m + 11$ for maximum parameter 7.

Established exact bounds for all such numbers with maximum parameter 7.

Provided new lower bounds for specific Folkman numbers, such as $F_v(2, 2, 2, 4; 5) \\geq 19$.

Abstract

For a graph $G$ the expression $G \overset{v}{\rightarrow} (a_1, ..., a_s)$ means that for every $s$-coloring of the vertices of $G$ there exists $i \in \{1, ..., s\}$ such that there is a monochromatic $a_i$-clique of color $i$. The vertex Folkman numbers $$F_v(a_1, ..., a_s; m - 1) = \min\{\vert V(G)\vert : G \overset{v}{\rightarrow} (a_1, ..., a_s) \mbox{ and } K_{m - 1} \not\subseteq G\}.$$ are considered, where $m = \sum_{i = 1}^{s}(a_i - 1) + 1$. We know the exact values of all the numbers $F_v(a_1, ..., a_s; m - 1)$ when $\max\{a_1, ..., a_s\} \leq 6$ and also the number $F_v(2, 2, 7; 8) = 20$. In \cite{BN15a} we present a method for obtaining lower bounds on these numbers. With the help of this method and a new improved algorithm, in the special case when $\max\{a_1, ..., a_s\} = 7$ we prove that $F_v(a_1, ..., a_s; m - 1) \geq m + 11$ and this bound is exact for all $m$. The known upper bound for these numbers is $m + 12$. At the end of the paper we also prove the lower bounds $19 \leq F_v(2, 2, 2, 4; 5)$ and $29 \leq F_v(7, 7; 8)$.