Around Erdős-Lovász problem on colorings of non-uniform hypergraphs

The talk deals with combinatorial problems concerning colorings of non-uniform hyper-graphs. Let H = (V, E) be a hypergraph with minimum edge-cardinality n. We show that if H is a simple hypergraph (i.e. every two distinct edges have at most one common vertex) and e∈E r 1−|e| c √ n, for some absolute constant c > 0, then H is r-colorable. We also obtain a stronger result for triangle-free simple hypergraphs by proving that if H is a simple triangle-free hypergraph and e∈E r 1−|e| c · n, for some absolute constant c > 0, then H is r-colorable.