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A note on perfect matchings in uniform hypergraphs with large minimum collective degree

Vojtěch Rödl;  Andrzej Ruciński;  Mathias Schacht;  Endre Szemerédi — 2008

Commentationes Mathematicae Universitatis Carolinae

For an integer k ≥ 2 and a k -uniform hypergraph H , let δ k - 1 ( H ) be the largest integer d such that every ( k - 1 ) -element set of vertices of H belongs to at least d edges of H . Further, let t ( k , n ) be the smallest integer t such that every k -uniform hypergraph on n vertices and with δ k - 1 ( H ) ≥ t contains a perfect matching. The parameter t ( k , n ) has been completely determined for all k and large n divisible by k by Rödl, Ruci’nski, and Szemerédi in [, submitted]. The values of t ( k , n ) are very close to n / 2 - k . In fact, the function t ( k , n ) = n / 2 - k + c n , k , where c n , k ∈ { 3 / 2 , 2 , 5 / 2 , 3 } depends...

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