Fractional and integer matchings in uniform hypergraphs

Our main result improves bounds of Markstrom and Rucinski on the minimum d-degree which forces a perfect matching in a k-uniform hypergraph on n vertices. We also extend bounds of Bollobas, Daykin and Erdos by asymptotically determining the minimum vertex degree which forces a matching of size t

[1]  Andrzej Rucinski,et al.  Perfect matchings (and Hamilton cycles) in hypergraphs with large degrees , 2011, Eur. J. Comb..

[2]  P. Erdös,et al.  Intersection Theorems for Systems of Sets , 1960 .

[3]  N. Sauer,et al.  On the factorization of the complete graph , 1973 .

[4]  D. Kuhn,et al.  Surveys in Combinatorics 2009: Embedding large subgraphs into dense graphs , 2009, 0901.3541.

[5]  Peter Keevash,et al.  Polynomial-time perfect matchings in dense hypergraphs☆ , 2013, 1307.2608.

[6]  Noga Alon,et al.  Nonnegative k-sums, fractional covers, and probability of small deviations , 2012, J. Comb. Theory, Ser. B.

[7]  David E. Daykin,et al.  Degrees giving independent edges in a hypergraph , 1981, Bulletin of the Australian Mathematical Society.

[8]  Vojtech Rödl,et al.  Perfect matchings in uniform hypergraphs with large minimum degree , 2006, Eur. J. Comb..

[9]  Fan Chung Graham,et al.  Regularity Lemmas for Hypergraphs and Quasi-randomness , 1991, Random Struct. Algorithms.

[10]  Ethan D. Bloch.  Infinite and Finite Sets , 2003 .

[11]  Vladimir Nikiforov,et al.  The number of cliques in graphs of given order and size , 2007, 0710.2305.

[12]  P. Erdos,et al.  On maximal paths and circuits of graphs , 1959 .

[13]  Vojtech Rödl,et al.  Large matchings in uniform hypergraphs and the conjectures of Erdős and Samuels , 2011, J. Comb. Theory, Ser. A.

[14]  J. Moon.  On Independent Complete Subgraphs in a Graph , 1968, Canadian Journal of Mathematics.

[15]  P. Erdös,et al.  INTERSECTION THEOREMS FOR SYSTEMS OF FINITE SETS , 1961 .

[16]  Peter Frankl,et al.  Improved bounds for Erdős' Matching Conjecture , 2013, J. Comb. Theory, Ser. A.

[17]  Yi Zhao,et al.  Exact minimum degree thresholds for perfect matchings in uniform hypergraphs , 2012, J. Comb. Theory, Ser. A.

[18]  Alexander A. Razborov,et al.  On the Minimal Density of Triangles in Graphs , 2008, Combinatorics, Probability and Computing.

[19]  P. Erdos.  A PROBLEM ON INDEPENDENT r-TUPLES , 1965 .

[20]  Hiêp Hàn,et al.  On Perfect Matchings in Uniform Hypergraphs with Large Minimum Vertex Degree , 2009, SIAM J. Discret. Math..

[21]  B. Andrásfai,et al.  Über ein Extremalproblem der Graphentheorie , 1962 .

[22]  E. Szemerédi.  Regular Partitions of Graphs , 1975 .

[23]  Hao Huang,et al.  The Size of a Hypergraph and its Matching Number , 2011, Combinatorics, Probability and Computing.

[24]  Vojtech Rödl,et al.  Perfect matchings in large uniform hypergraphs with large minimum collective degree , 2009, J. Comb. Theory, Ser. A.

[25]  Imdadullah Khan,et al.  Perfect Matchings in 3-Uniform Hypergraphs with Large Vertex Degree , 2011, SIAM J. Discret. Math..

[26]  Peter Frankl,et al.  Improved bounds for Erd ˝ os' Matching Conjecture , 2013 .

[27]  P. Erdös.  über ein Extremalproblem in der Graphentheorie , 1962 .

[28]  Daniela Kühn,et al.  Matchings in 3-uniform hypergraphs , 2010, J. Comb. Theory, Ser. B.

[29]  Imdadullah Khan,et al.  Perfect matchings in 4-uniform hypergraphs , 2011, J. Comb. Theory, Ser. B.

[30]  M. Simonovits,et al.  On the number of complete subgraphs of a graph II , 1983 .

[31]  Daniela Kühn,et al.  Matchings in hypergraphs of large minimum degree , 2006 .

[32]  Vojtech Rödl,et al.  On the Maximum Number of Edges in a Triple System Not Containing a Disjoint Family of a Given Size , 2012, Combinatorics, Probability and Computing.

[33]  D. E. Daykin,et al.  SETS OF INDEPENDENT EDGES OF A HYPERGRAPH , 1976 .

[34]  B. Bollobás.  Surveys in Combinatorics , 1979 .

[35]  Oleg Pikhurko,et al.  Perfect Matchings and K43-Tilings in Hypergraphs of Large Codegree , 2008, Graphs Comb..

[36]  TOMASZ LUCZAK,et al.  On Erdős' extremal problem on matchings in hypergraphs , 2012, J. Comb. Theory, Ser. A.

[37]  Christian Reiher,et al.  The clique density theorem , 2012, 1212.2454.