Almost all Steiner triple systems are almost resolvable

Abstract We show that for any n divisible by 3, almost all order-n Steiner triple systems admit a decomposition of almost all their triples into disjoint perfect matchings (that is, almost all Steiner triple systems are almost resolvable).

[1]  Daniela Kühn,et al.  Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments , 2012, ArXiv.

[2]  D. K. Ray-Chaudhuri,et al.  Solution of Kirkman''s schoolgirl problem , 1971 .

[3]  Peter Keevash,et al.  The existence of designs II , 2018, 1802.05900.

[4]  Yoshiharu Kohayakawa,et al.  OnK4-free subgraphs of random graphs , 1997, Comb..

[5]  Asaf Ferber,et al.  Packing, counting and covering Hamilton cycles in random directed graphs , 2017 .

[6]  Peter Keevash,et al.  A Geometric Theory for Hypergraph Matching , 2011, 1108.1757.

[7]  Alan M. Frieze,et al.  Packing hamilton cycles in random and pseudo-random hypergraphs , 2012, Random Struct. Algorithms.

[8]  Asaf Ferber,et al.  Packing, counting and covering Hamilton cycles in random directed graphs , 2015, Electron. Notes Discret. Math..

[9]  Matthew Kwan Almost all Steiner triple systems have perfect matchings , 2016, Proceedings of the London Mathematical Society.

[10]  Svante Janson,et al.  Random graphs , 2000, ZOR Methods Model. Oper. Res..

[11]  Benny Sudakov,et al.  Random regular graphs of high degree , 2001, Random Struct. Algorithms.

[12]  D. Freedman On Tail Probabilities for Martingales , 1975 .

[13]  Arthur Cayley,et al.  The Collected Mathematical Papers: On the Triadic Arrangements of Seven and Fifteen Things , 1850 .

[14]  Daniela Kühn,et al.  Edge‐disjoint Hamilton cycles in random graphs , 2011, Random Struct. Algorithms.

[15]  Brendan D. McKay,et al.  Most Latin Squares Have Many Subsquares , 1999, J. Comb. Theory A.

[16]  Prince Camille de Polignac On a Problem in Combinations , 1866 .

[17]  W. T. Gowers,et al.  On the KŁR conjecture in random graphs , 2013, 1305.2516.

[18]  David Conlon,et al.  Weak quasi‐randomness for uniform hypergraphs , 2012, Random Struct. Algorithms.

[19]  R. Quackenbush Algebraic Speculations About Steiner Systems , 1980 .

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

[21]  Vojtech Rödl,et al.  An approximate Dirac-type theorem for k-uniform hypergraphs , 2008, Comb..

[22]  Packing , 2020, Definitions.

[23]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[24]  Gábor Lugosi,et al.  Concentration Inequalities - A Nonasymptotic Theory of Independence , 2013, Concentration Inequalities.

[25]  Yoshiharu Kohayakawa,et al.  Szemerédi’s Regularity Lemma and Quasi-randomness , 2003 .

[26]  Yoshiharu Kohayakawa,et al.  Weak hypergraph regularity and linear hypergraphs , 2010, J. Comb. Theory, Ser. B.

[27]  Svante Janson,et al.  Random graphs , 2000, Wiley-Interscience series in discrete mathematics and optimization.

[28]  Peter Keevash The existence of designs , 2014, 1401.3665.

[29]  László Babai Almost All Steiner Triple Systems Are Asymmetric , 1980 .

[30]  Peter J. Cameron A generalisation of t-designs , 2009, Discret. Math..

[31]  Jeff Kahn,et al.  Factors in random graphs , 2008, Random Struct. Algorithms.

[32]  T. Lu ON K4-FREE SUBGRAPHS OF RANDOM GRAPHS , 1997 .

[33]  Endre Szemer,et al.  AN APPROXIMATE DIRAC-TYPE THEOREM FOR k-UNIFORM HYPERGRAPHS , 2008 .

[34]  Svante Janson,et al.  The infamous upper tail , 2002, Random Struct. Algorithms.

[35]  N. Linial,et al.  Discrepancy of High-Dimensional Permutations , 2015, 1512.04123.

[36]  Benny Sudakov,et al.  Intercalates and discrepancy in random Latin squares , 2018, Random Struct. Algorithms.

[37]  R. Sarpong,et al.  Bio-inspired synthesis of xishacorenes A, B, and C, and a new congener from fuscol† †Electronic supplementary information (ESI) available. See DOI: 10.1039/c9sc02572c , 2019, Chemical science.

[38]  R. Montgomery Embedding bounded degree spanning trees in random graphs , 2014, 1405.6559.

[39]  Vojtech Rödl,et al.  On a Packing and Covering Problem , 1985, Eur. J. Comb..

[40]  Joel H. Spencer,et al.  Asymptotic behavior of the chromatic index for hypergraphs , 1989, J. Comb. Theory, Ser. A.

[41]  Peter Keevash Counting designs , 2015 .

[42]  Darryn E. Bryant,et al.  Steiner Triple Systems without Parallel Classes , 2015, SIAM J. Discret. Math..

[43]  Jeong Han Kim,et al.  Nearly perfect matchings in regular simple hypergraphs , 1997 .

[44]  Ian M. Wanless,et al.  The Existence of Latin Squares without Orthogonal Mates , 2006, Des. Codes Cryptogr..

[45]  Lutz Warnke,et al.  On the Method of Typical Bounded Differences , 2012, Combinatorics, Probability and Computing.