The size of a hypergraph and its matching number

Details The size of a hypergraph and its matching number The size of a hypergraph and its matching number

2011-07-27

arxiv.org/abs/1107.5544v2

Mathematics

Combinatorics

9 pages

Scientific paper

More than forty years ago, Erd\H{o}s conjectured that for any T <= N/K, every K-uniform hypergraph on N vertices without T disjoint edges has at most max{\binom{KT-1}{K}, \binom{N}{K} - \binom{N-T+1}{K}} edges. Although this appears to be a basic instance of the hypergraph Tur\'an problem (with a T-edge matching as the excluded hypergraph), progress on this question has remained elusive. In this paper, we verify this conjecture for all T < N/(3K^2). This improves upon the best previously known range T = O(N/K^3), which dates back to the 1970's.

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