Maximum degree and fractional matchings in uniform hypergraphs

Abstract

Let ℋ be a family ofr-subsets of a finite setX. SetD()=\(\mathop {\max }\limits_{x \in X} \)|{E:xE}|, (maximum degree). We say that ℋ is intersecting if for anyH,H′ ∈ ℋ we haveHH′ ≠ 0. In this case, obviously,D(ℋ)≧|ℋ|/r. According to a well-known conjectureD(ℋ)≧|ℋ|/(r−1+1/r). We prove a slightly stronger result. Let ℋ be anr-uniform, intersecting hypergraph. Then either it is a projective plane of orderr−1, consequentlyD(ℋ)=|ℋ|/(r−1+1/r), orD(ℋ)≧|ℋ|/(r−1). This is a corollary to a more general theorem on not necessarily intersecting hypergraphs.

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Füredi, Z. Maximum degree and fractional matchings in uniform hypergraphs. Combinatorica 1, 155–162 (1981). https://doi.org/10.1007/BF02579271

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AMS subject classification (1980)

  • 05 C 65, 05 C 35
  • 05 B 25