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Erdős–Ko–Rado for Random Hypergraphs: Asymptotics and Stability

Part of: Extremal combinatorics

Published online by Cambridge University Press:  29 March 2017

MARCELO M. GAUY ,
HIÊP HÀN  and
IGOR C. OLIVEIRA
MARCELO M. GAUY
Affiliation:
Institute of Theoretical Computer Science, ETH Zürich, 8092 Zürich, Switzerland
HIÊP HÀN
Affiliation:
Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, Blanco Viel 596, Cerro Barón, Valparaíso, Chile
IGOR C. OLIVEIRA
Affiliation:
Faculty of Mathematics and Physics, Charles University in Prague, Sokolovska 83, 186 75 Prague 8, Czech Republic
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Abstract

We investigate the asymptotic version of the Erdős–Ko–Rado theorem for the random k-uniform hypergraph $\mathcal{H}$ k (n, p). For 2⩽k(n) ⩽ n/2, let $N=\binom{n}k$ and $D=\binom{n-k}k$ . We show that with probability tending to 1 as n → ∞, the largest intersecting subhypergraph of $\mathcal{H}$ has size

$$(1+o(1))p\ffrac kn N$$
for any
$$p\gg \ffrac nk\ln^2\biggl(\ffrac nk\biggr)D^{-1}.$$
This lower bound on p is asymptotically best possible for k = Θ(n). For this range of k and p, we are able to show stability as well.

A different behaviour occurs when k = o(n). In this case, the lower bound on p is almost optimal. Further, for the small interval D −1 ≪ p ⩽ (n/k)1−ϵ D −1, the largest intersecting subhypergraph of $\mathcal{H}$ k (n, p) has size Θ(ln(pD)ND −1), provided that $k \gg \sqrt{n \ln n}$ .

Together with previous work of Balogh, Bohman and Mubayi, these results settle the asymptotic size of the largest intersecting family in $\mathcal{H}$ k , for essentially all values of p and k.

MSC classification

Primary: 05D05: Extremal set theory
Secondary: 05D40: Probabilistic methods

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 3 , May 2017 , pp. 406 - 422
Copyright
Copyright © Cambridge University Press 2017 

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References

[1] Ajtai, M., Komlós, J. and Szemerédi, E. (1980) A note on Ramsey numbers, J. Combin. Theory Ser. A 29 354–360.Google Scholar
[2] Alon, N., Dinur, I., Friedgut, E. and Sudakov, B. (2004) Graph products, Fourier analysis and spectral techniques. Geom. Funct. Anal. 14 913–940.Google Scholar
[3] Balogh, J., Bohman, T. and Mubayi, D. (2009) Erdős–Ko–Rado in random hypergraphs. Combin. Probab. Comput. 18 629–646.CrossRefGoogle Scholar
[4] Balogh, J., Bollobás, B. and Narayanan, B. Transference for the Erdős–Ko–Rado theorem. Forum of Math. Sigma, to appear https://doi.org/10.1017/fms.2015.21.Google Scholar
[5] Balogh, J., Das, S., Delcourt, M., Liu, H. and Sharifzadeh, M. (2015) Intersecting families of discrete structures are typically trivial. J. Combin. Theory Ser. A 132 224–245.Google Scholar
[6] Balogh, J., Morris, R. and Samotij, W. (2015) Independent sets in hypergraphs. J. Amer. Math. Soc. 28 669–709.Google Scholar
[7] Bollobás, B., Narayanan, B. and Raigorodskii, A. On the stability of the Erdős–Ko–Rado theorem. J. Combin. Theory Ser. A, to appear http://dx.doi.org/10.1016/j.jcta.2015.08.002.Google Scholar
[8] Conlon, D. and Gowers, W. Combinatorial theorems in sparse random sets. Submitted DOI 10.4007/annals.2016.184.2.2.Google Scholar
[9] Das, S. and Tran, T. Removal and stability for Erdős–Ko–Rado. SIAM J. Discrete Math., to appear DOI:10.1137/15M105149X.Google Scholar
[10] Devlin, P. and Kahn, J. On stability in the Erdős–Ko–Rado theorem. Submitted DOI:10.1137/15M1012992.Google Scholar
[11] Erdős, P., Ko, C. and Rado, R. (1961) Intersection theorems for systems of finite sets. Quart. J. Math. 12 313–320.Google Scholar
[12] Friedgut, E. (2008) On the measure of intersecting families, uniqueness and stability. Combinatorica 28 503–528.Google Scholar
[13] Friedgut, E. and Regev, O. Manuscript.Google Scholar
[14] Hamm, A. and Kahn, J. On Erdős–Ko–Rado for random hypergraphs I. Submitted.Google Scholar
[15] Hamm, A. and Kahn, J. On Erdős–Ko–Rado for random hypergraphs II. Submitted.Google Scholar
[16] Hoffman, A. (1970) On eigenvalues and colorings of graphs. In Graph Theory and its Applications (Proc. Advanced Sem., Math. Research Center, Univ. of Wisconsin), pp. 79–91.Google Scholar
[17] Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience.Google Scholar
[18] Kleitman, D. and Winston, K. (1982) On the number of graphs without 4-cycles. Discrete Math. 41 167–172.Google Scholar
[19] Kohayakawa, Y., Kreuter, B. and Steger, A. (1998) An extremal problem for random graphs and the number of graphs with large even-girth. Combinatorica 18 101–120.Google Scholar
[20] Kohayakawa, Y., Lee, S. J., Rödl, V. and Samotij, W. (2015) The number of Sidon sets and the maximum size of Sidon sets contained in a sparse random set of integers. Random Struct. Alg. 46 1–25.Google Scholar
[21] Lovász, L. (1979) On the Shannon capacity of a graph. IEEE Trans. Inform. Theory 25 1–7.Google Scholar
[22] Samotij, W. (2014) Stability results for random discrete structures. Random Struct. Alg. 44 269–289.Google Scholar
[23] Saxton, D. and Thomason, A. (2015) Hypergraph containers. Invent. Math. 201 925–992.Google Scholar
[24] Schacht, M. Extremal results for random discrete structures. Submitted DOI 10.4007/annals.2016.184.2.1.Google Scholar
[25] Shearer, J. (1983) A note on the independence number of triangle-free graphs. Discrete Math. 46 83–87.Google Scholar