A Folkman Linear Family

TL;DR

This paper establishes an upper bound on Folkman numbers for graphs with bounded maximum degree, showing they grow linearly with the order of the graph, which advances understanding of graph Ramsey properties.

Contribution

The paper proves that Folkman numbers for graphs with bounded maximum degree are linearly bounded in the size of the graph, providing new bounds in graph Ramsey theory.

Findings

Folkman numbers grow linearly with graph size for bounded degree graphs.

Established bounds depend only on maximum degree, not on graph size.

Provides explicit constants for the bounds based on maximum degree.

Abstract

For graphs $F$ and $G$, let $F\to (G,G)$ signify that any red/blue edge coloring of $F$ contains a monochromatic $G$. Define Folkman number $f(G;p)$ to be the smallest order of a graph $F$ such that $F\to (G,G)$ and $\omega(F) \le p$. It is shown that $f(G;p)\le cn$ for graphs $G$ of order $n$ with $\Delta(G)\le \Delta$, where $\Delta\ge 3$, $c=c(\Delta)$ and $p=p(\Delta)$ are positive constants.