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Forbidden Hypermatrices Imply General Bounds on Induced Forbidden Subposet Problems

  • ABHISHEK METHUKU (a1) and DÖMÖTÖR PÁLVÖLGYI (a2)
Abstract

We prove that for every poset P, there is a constant CP such that the size of any family of subsets of {1, 2, . . ., n} that does not contain P as an induced subposet is at most $$C_P{\binom{n}{\lfloor\gfrac{n}{2}\rfloor}},$$ settling a conjecture of Katona, and Lu and Milans. We obtain this bound by establishing a connection to the theory of forbidden submatrices and then applying a higher-dimensional variant of the Marcus–Tardos theorem, proved by Klazar and Marcus. We also give a new proof of their result.

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[1] Boehnlein E. and Jiang T. (2012) Set families with a forbidden induced subposet. Combin. Probab. Comput. 21 496511.
[2] Bukh B. (2009) Set families with a forbidden subposet. Electron. J. Combin. 16 R142.
[3] Burcsi P. and Nagy D. T. (2013) The method of double chains for largest families with excluded subposet. Electron. J. Graph Theory Appl. 1 4049.
[4] Carroll T. and Katona G. O. H. (2008) Bounds on maximal families of sets not containing three sets with ABC, A ¬ ⊂ B. Order 25 229236.
[5] Chen H. B. and Li W.-T. (2014) A note on the largest size of families of sets with a forbidden poset. Order 31 137142.
[6] Erdős P. (1945) On a lemma of Littlewood and Offord. Bull. Amer. Math. Soc. 51 898902.
[7] Fox J. (2013) Stanley–Wilf limits are typically exponential. arXiv:1310.8378
[8] Füredi Z. and Hajnal P. (1992) Davenport–Schinzel theory of matrices. Discrete Math. 103 233251.
[9] Geneson J. T. and Tian P. M. (2015) Extremal functions of forbidden multidimensional matrices. arXiv:1506.03874
[10] Griggs J. R. and Li W.-T. (2016) Progress on poset-free families of subsets. In Recent Trends in Combinatorics (Beveridge A. et al., eds), Springer, pp. 317338.
[11] Griggs J. R., Li W.-T. and Lu L. (2012) Diamond-free families. J. Combin. Theory. Ser. A 119 310322.
[12] Griggs J. R. and Lu L. (2009) On families of subsets with a forbidden subposet. Combin. Probab. Comput. 18 731748.
[13] Grósz D., Methuku A. and Tompkins C. (2014) An improvement of the general bound on the largest family of subsets avoiding a subposet. Order, pp. 1–13.
[14] Grósz D., Methuku A. and Tompkins C. (2016) An upper bound on the size of diamond-free families of sets. arXiv:1601.06332
[15] Katona G. O. H. (1972) A simple proof of the Erdős–Chao Ko–Rado theorem. J. Combin. Theory. Ser. B 13 183184.
[16] Katona G. O. H. (2008) Forbidden intersection patterns in the families of subsets (introducing a method). In Horizons of Combinatorics (Győri E., Katona G. and Lovász L., eds), Springer, pp. 119140.
[17] Katona G. O. H. (2012) Personal communication.
[18] Katona G. O. H. and Tarján T. G. (1983) Extremal problems with excluded subgraphs in the n-cube. In Graph Theory (Borowiecki M., Kennedy J. W. and Sysło M. M., eds), Springer, pp. 8493.
[19] Klazar M. and Marcus A. (2006) Extensions of the linear bound in the Füredi–Hajnal conjecture. Adv. Appl. Math. 38 258266.
[20] Loomis H. L. and Whitney H. (1949) An inequality related to the isoperimetric inequality. Bull. Amer. Math. Soc. 55 961962.
[21] Lu L. and Milans K. G. (2015) Set families with forbidden subposets. Journal of Combinatorial Theory, Series A 136 126142.
[22] Marcus A. and Tardos G. (2004) Excluded permutation matrices and the Stanley–Wilf conjecture. J. Combin. Theory. Ser. A 107 153160.
[23] Méroueh A. (2015) Lubell mass and induced partially ordered sets. arXiv:1506.07056
[24] Patkós B. (2015) Induced and non-induced forbidden subposet problems. Electron. J. Combin. 22 P1.30.
[25] Sperner E. (1928) Ein Satz über Untermegen einer endlichen Menge. Math. Z. 27 544548.
[26] Tardos G. (2005) On 0–1 matrices and small excluded submatrices. J. Combin. Theory. Ser. A 111 266288.
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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