A Note on Upper Bounds for Some Generalized Folkman Numbers

TL;DR

This paper introduces new constructive upper bounds for generalized Folkman numbers using product graphs, improving known bounds for specific cases and providing explicit graph constructions to demonstrate these bounds.

Contribution

It presents novel upper bounds for generalized vertex and edge Folkman numbers through explicit graph constructions based on product graphs.

Findings

F_e(K_3,K_4-e; K_5)   27

F_e(K_4-e,K_4-e; K_5)   51

Constructed a 51-vertex K_5-free graph with specific coloring properties

Abstract

We present some new constructive upper bounds based on product graphs for generalized vertex Folkman numbers. They lead to new upper bounds for some special cases of generalized edge Folkman numbers, including $F_e(K_3,K_4-e; K_5) \leq 27$ and $F_e(K_4-e,K_4-e; K_5) \leq 51$. The latter bound follows from a construction of a $K_5$-free graph on 51 vertices, for which every coloring of its edges with two colors contains a monochromatic $K_4-e$.