For fixed positive integers $r, k$ and $ell$ with $1 leq ell < r$ and an $r$-uniform hypergraph $H$, let $kappa (H, k,ell)$ denote the number of $k$-colorings of the set of hyperedges of $H$ for which any two hyperedges in the same color class inters
ect in at least $ell$ elements. Consider the function $KC(n,r,k,ell)=max_{Hin{mathcal H}_{n}} kappa (H, k,ell) $, where the maximum runs over the family ${mathcal H}_n$ of all $r$-uniform hypergraphs on $n$ vertices. In this paper, we determine the asymptotic behavior of the function $KC(n,r,k,ell)$ for every fixed $r$, $k$ and $ell$ and describe the extremal hypergraphs. This variant of a problem of ErdH{o}s and Rothschild, who considered edge colorings of graphs without a monochromatic triangle, is related to the ErdH{o}s--Ko--Rado Theorem on intersecting systems of sets [Intersection Theorems for Systems of Finite Sets, Quarterly Journal of Mathematics, Oxford Series, Series 2, {bf 12} (1961), 313--320].
An induced forest of a graph G is an acyclic induced subgraph of G. The present paper is devoted to the analysis of a simple randomised algorithm that grows an induced forest in a regular graph. The expected size of the forest it outputs provides a l
ower bound on the maximum number of vertices in an induced forest of G. When the girth is large and the degree is at least 4, our bound coincides with the best bound known to hold asymptotically almost surely for random regular graphs. This results in an alternative proof for the random case.
