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.

It is shown by using differential methods that if ${\cal H}$ is a double linear, r-uniform hypergraph with degree sequence $\{d_v\}$ such that any subhypergraph induced by a neighborhood has maximum degree less than m, then its independence number is at least $\sum_{v}f_{r,m}(d_v)$, where $f_{r,m}(x)$ is a convex function satisfying $f_{r,m}(x)\sim (\log x)/x$ if $r=2$ and $c/x^{1/(r-1)}$ if $r \ge 3$, as $x\to\infty$, and $c=c(r,m)>0$ is a constant. The proof yields a polynomial-time algorithm for finding such an independent set in ${\cal H}$.