On the Vertex Folkman Numbers $F_v(2,...,2;q)$

TL;DR

This paper computes specific vertex Folkman numbers for certain parameters and establishes new bounds for some Folkman numbers, advancing understanding in graph coloring and Ramsey theory.

Contribution

It provides exact values of vertex Folkman numbers $F_v(2,...,2;r-k+1)$ for $k \\le 12$ and large $r$, and introduces new bounds for related Folkman numbers.

Findings

Computed Folkman numbers for $k \\le 12$ and large $r$

Established new bounds for vertex Folkman numbers

Enhanced understanding of graph colorings and Ramsey properties

Abstract

For a graph $G$ the symbol $G\tov(a_1,...,a_r)$ means that in every $r$-coloring of the vertices of $G$ for some $i\in\{1,...,r\}$ there exists a monochromatic $a_i$-clique of color $i$. The vertex Folkman numbers \[ \FN=\min\{|V(G)|:G\tov(a_1,...,a_r)\text{and}K_q\nsubseteqq G\} \] are considered. In this paper we shall compute the Folkman numbers $F_v(\underbrace{2,...,2}_r;r-k+1)$ when $k\le 12$ and $r$ is sufficiently large. We prove also new bounds for some vertex and edge Folkman numbers.