The clique complex and hypergraph matching

The width of a hypergraph F is the minimal t = w(F) for which there exist F-1,..., F-t is an element of F such that for any F is an element of F, F-i boolean AND F not equal circle divide for some 1 less than or equal to i less than or equal to t. The matching width of F is the minimal t = mw(F) such that for any matching M subset of F there exist F-1,..., F-t is an element of F such that for any F is an element of M, F(i)boolean ANDF not equal circle divide for some 1 less than or equal to i less than or equal to t. The following extension of the Aharoni-Haxell matching Theorem [3] is proved: Let {F-i}(i = 1)(m) be a family of hypergraphs such that for each circle divide not equal I subset of [m] either mw(Ui is an element ofIFi) greater than or equal to \I\ or w(Ui is an element ofIFi) greater than or equal to 2\I\ -1, then there exists a matching F-1,...,F-m such that F-i is an element of F-i for all 1 less than or equal to i less than or equal to m. This is a consequence of a more general result on colored cliques in graphs. The proofs are topological and use the Nerve Theorem.