Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-25T17:50:19.836Z Has data issue: false hasContentIssue false
  • English
  • Français

Forbidden Hypermatrices Imply General Bounds on Induced Forbidden Subposet Problems

Published online by Cambridge University Press:  15 February 2017

ABHISHEK METHUKU  and
DÖMÖTÖR PÁLVÖLGYI
ABHISHEK METHUKU
Affiliation:
Department of Mathematics, Central European University, 1051 Budapest, Hungary (e-mail: [email protected])
DÖMÖTÖR PÁLVÖLGYI
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WA, UK (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

Primary 05D05Secondary 05A05
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 4 , July 2017 , pp. 593 - 602
Copyright
Copyright © Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Boehnlein, E. and Jiang, T. (2012) Set families with a forbidden induced subposet. Combin. Probab. Comput. 21 496–511.Google Scholar
[2] Bukh, B. (2009) Set families with a forbidden subposet. Electron. J. Combin. 16 R142.Google Scholar
[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 40–49.Google Scholar
[4] Carroll, T. and Katona, G. O. H. (2008) Bounds on maximal families of sets not containing three sets with A ∪ B ⊂ C, A ¬ ⊂ B. Order 25 229–236.Google Scholar
[5] Chen, H. B. and Li, W.-T. (2014) A note on the largest size of families of sets with a forbidden poset. Order 31 137–142.CrossRefGoogle Scholar
[6] Erdős, P. (1945) On a lemma of Littlewood and Offord. Bull. Amer. Math. Soc. 51 898–902.CrossRefGoogle Scholar
[7] Fox, J. (2013) Stanley–Wilf limits are typically exponential. arXiv:1310.8378 Google Scholar
[8] Füredi, Z. and Hajnal, P. (1992) Davenport–Schinzel theory of matrices. Discrete Math. 103 233–251.Google Scholar
[9] Geneson, J. T. and Tian, P. M. (2015) Extremal functions of forbidden multidimensional matrices. arXiv:1506.03874 Google Scholar
[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. 317–338.Google Scholar
[11] Griggs, J. R., Li, W.-T. and Lu, L. (2012) Diamond-free families. J. Combin. Theory. Ser. A 119 310–322.Google Scholar
[12] Griggs, J. R. and Lu, L. (2009) On families of subsets with a forbidden subposet. Combin. Probab. Comput. 18 731–748.Google Scholar
[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.Google Scholar
[14] Grósz, D., Methuku, A. and Tompkins, C. (2016) An upper bound on the size of diamond-free families of sets. arXiv:1601.06332 Google Scholar
[15] Katona, G. O. H. (1972) A simple proof of the Erdős–Chao Ko–Rado theorem. J. Combin. Theory. Ser. B 13 183–184.Google Scholar
[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. 119–140.Google Scholar
[17] Katona, G. O. H. (2012) Personal communication.Google Scholar
[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. 84–93.Google Scholar
[19] Klazar, M. and Marcus, A. (2006) Extensions of the linear bound in the Füredi–Hajnal conjecture. Adv. Appl. Math. 38 258–266.Google Scholar
[20] Loomis, H. L. and Whitney, H. (1949) An inequality related to the isoperimetric inequality. Bull. Amer. Math. Soc. 55 961–962.Google Scholar
[21] Lu, L. and Milans, K. G. (2015) Set families with forbidden subposets. Journal of Combinatorial Theory, Series A 136 126–142.Google Scholar
[22] Marcus, A. and Tardos, G. (2004) Excluded permutation matrices and the Stanley–Wilf conjecture. J. Combin. Theory. Ser. A 107 153–160.Google Scholar
[23] Méroueh, A. (2015) Lubell mass and induced partially ordered sets. arXiv:1506.07056 Google Scholar
[24] Patkós, B. (2015) Induced and non-induced forbidden subposet problems. Electron. J. Combin. 22 P1.30.Google Scholar
[25] Sperner, E. (1928) Ein Satz über Untermegen einer endlichen Menge. Math. Z. 27 544–548.CrossRefGoogle Scholar
[26] Tardos, G. (2005) On 0–1 matrices and small excluded submatrices. J. Combin. Theory. Ser. A 111 266–288.Google Scholar