Random coloring method in the combinatorial problem of Erdős and Lovász
Random Structures & Algorithms
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A hypergraph is b-simple if no two distinct edges share more than b vertices. Let m(r, t, g) denote the minimum number of edges in an r-uniform non-t-colorable hypergraph of girth at least g. Erdős and Lovász proved that $$\eqalign { & \qquad m(r,t,3)\geq {t^{2(r-2)} \over 16r(r-1)^2} \cr & {\hbox {and}} \quad m(r,t,g)\leq 4\cdot 20^{g-1} r^{3g-5} t^{(g-1)(r+1)}.}$$ A result of Szabó improves the lower bound by a factor of r2-&epsis; for sufficiently large r. We improve the lower bound by another factor of r and extend the result to b-simple hypergraphs. We also get a new lower bound for hypergraphs with a given girth. Our results imply that for fixed b, t, and &epsis; 0 and sufficiently large r, every r-uniform b-simple hypergraph $\cal {H}$ with maximum edge-degree at most trr1-&epsis; is t-colorable. Some results hold for list coloring, as well. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009