On the vertex Folkman numbers $F_v(a_1, ..., a_s; m - 1)$ when $\max\{a_1, ..., a_s\} = 6$ or $7$

TL;DR

This paper computes specific vertex Folkman numbers for graphs with clique constraints when the maximum of the parameters is 6, filling a gap in the existing knowledge for these combinatorial invariants.

Contribution

The paper determines the values of vertex Folkman numbers $F_v(a_1, ..., a_s; m - 1)$ for cases where the maximum parameter is 6, extending previous results known only for maximum 5.

Findings

Computed $F_v(a_1, ..., a_s; m - 1)$ for maximum parameter 6.

Extended the known range of Folkman numbers beyond previous limits.

Provided new exact values for specific graph coloring and clique avoidance scenarios.

Abstract

Let $G$ be a graph and $a_1, ..., a_s$ be positive integers. Then $G \overset{v}{\rightarrow} (a_1, ..., a_s)$ 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 by the equality: $$ F_v(a_1, ..., a_s; q) = \min\{|V(G)| : G \overset{v}{\rightarrow} (a_1, ..., a_s) \mbox{ and } K_q \not\subseteq G\}. $$ Let $m = \sum\limits_{i=1}^s (a_i - 1) + 1$. It is easy to see that $F_v(a_1, ..., a_s; q) = m$ if $q \geq m + 1$. In [11] it is proved that $F_v(a_1, ..., a_s; m) = m + \max\{a_1, ..., a_s\}$. We know all the numbers $F_v(a_1, ..., a_s; m - 1)$ when $\max\{a_1, ..., a_s\} \leq 5$ and none of these numbers is known if $\max\{a_1, ..., a_s\} \geq 6$. In this paper we compute the numbers $F_v(a_1, ..., a_s; m - 1)$ when $\max\{a_1, ..., a_s\} = 6$.