A Folkman Linear Family

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 $ω(F) \le p$. It is shown that $f(G;p)\le cn$ for graphs $G$ of order $n$ with $Δ(G)\le Δ$, where $Δ\ge 3$, $c=c(Δ)$ and $p=p(Δ)$ are positive constants.

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