Sperner theory
Improved bounds and algorithms for hypergraph 2-coloring
Random Structures & Algorithms
Color critical hypergraphs with many edges
Journal of Graph Theory
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A hypergraph ([n],E) is 3-color critical if it is not 2-colorable, but for all E@?E the hypergraph ([n],E@?{E}) is 2-colorable. Lovasz proved in 1976, that |E|@?nk-1 if E is k-uniform. Here we give a new algebraic proof and an ordered version that is a sharpening of Lovasz' result. Let E@?[n]k be a k-uniform set system on an underlying set [n] of n elements. Let us fix an ordering E"1,E"2,...E"t of E and a prescribed partition {A"i,B"i} of each E"i (i.e., A"i@?B"i=E"i and A"i@?B"i=0@?). Assume that for all i=1,2,...,t there exists a partition {C"i,D"i} of [n] such that E"i@?C"i=A"i and E"i@?D"i=B"i, but {E"j@?C"i,E"j@?D"i}{A"j,B"j} for all j~). We also give constructions of sizes nk-1 for all n and k. Furthermore, in the 3-color-critical case (i.e. {A"i,B"i}={E"i,0@?} for all i), t@?nk-1.