Skip to main content Accessibility help
×
Home

On Erdős–Ko–Rado for Random Hypergraphs II

  • A. HAMM (a1) and J. KAHN (a2)

Abstract

Denote by ${\mathcal H}_k$ (n, p) the random k-graph in which each k-subset of {1,. . .,n} is present with probability p, independent of other choices. More or less answering a question of Balogh, Bohman and Mubayi, we show: there is a fixed ε > 0 such that if n = 2k + 1 and p > 1 - ε, then w.h.p. (that is, with probability tending to 1 as k → ∞), ${\mathcal H}_k$ (n, p) has the ‘Erdős–Ko–Rado property’. We also mention a similar random version of Sperner's theorem.

Copyright

Footnotes

Hide All

Supported by NSF grant DMS1201337.

Footnotes

References

Hide All
[1] Alon, N. and Spencer, J. H. (2008) The Probabilistic Method, Wiley.
[2] Babai, L., Simonovits, M. and Spencer, J. (1990) Extremal subgraphs of random graphs. J. Graph Theory 14 599622.
[3] Balogh, J. (2014) On the applications of counting independent sets in hypergraphs. In Workshop on Probabilistic and Extremal Combinatorics, IMA.
[4] Balogh, J., Bohman, T. and Mubayi, D. (2009) Erdős–Ko–Rado in random hypergraphs. Combin. Probab. Comput. 18 629646.
[5] Balogh, J., Morris, R. and Samotij, W. (2015) Independent sets in hypergraphs. J. Amer. Math. Soc. 28 669709.
[6] Balogh, J., Mycroft, R. and Treglown, A. (2014) A random version of Sperner's theorem. J. Combin. Theory Ser. A 128 104110.
[7] Bollobás, B. (1986) Combinatorics, Cambridge University Press.
[8] Collares~Neto, M. and Morris, R. (2016) Maximum-size antichains in random set-systems. Random Struct. Alg. 49 308321.
[9] Conlon, D. and Gowers, T. (2016) Combinatorial theorems in sparse random sets. Ann. Math. 184 367454.
[10] DeMarco, R. and Turán's, Kahn, J. Theorem for random graphs. arXiv1501.01340v1
[11] Erdős, P., Ko, C. and Rado, R. (1961) Intersection theorems for systems of finite sets. Quart. J. Math. Oxford Ser. (2) 12 313320.
[12] Frankl, P. (1987) Erdős–Ko–Rado theorem with conditions on the maximal degree. J. Combin. Theory Ser. A 46 252263.
[13] Frankl, P. and Rödl, V. (1986) Large triangle-free subgraphs in graphs without K 4. Graphs Combin. 2 135144.
[14] Hamm, A. and Kahn, J. On Erdős–Ko–Rado for random hypergraphs I. arXiv1412.5085v1
[15] Hilton, A. J. W. and Milner, E. C. (1967) Some intersection theorems for systems of finite sets. Quart. J. Math. Oxford Ser. (2) 18 369384.
[16] Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley.
[17] Katona, G. O. H. (1966) A theorem on finite sets. In Theory of Graphs: Proc. Colloq. Tihany, Akademiai Kiado / Academic Press, pp. 187207.
[18] Knuth, D. E. (1969) The Art of Computer Programming, Vol. I, Addison Wesley.
[19] Kohayakawa, Y., Łuczak, T. and Rödl, V. (1997) On K 4-free subgraphs of random graphs. Combinatorica 17 173213.
[20] Kruskal, J. B. (1963) The number of simplices in a complex. In Mathematical Optimization Techniques (Bellman, R., ed.), University of California Press, pp. 251278.
[21] Lovász, L. (1993) Combinatorial Problems and Exercises, American Mathematical Society.
[22] Osthus, D. (2000) Maximum antichains in random subsets of a finite set. J. Combin. Theory Ser. A 90 336346.
[23] Ramsey, F. B. (1930) On a problem of formal logic. Proc. London Math. Soc. 30 264286.
[24] Rödl, V. and Ruciński, A. (1995) Threshold functions for Ramsey properties. J. Amer. Math. Soc. 8 917942.
[25] Rödl, V. and Schacht, M. (2013) Extremal results in random graphs. In Erdős Centennial (Lovász, L. et al. eds), Vol. 25 of Bolyai Society Mathematical Studies, János Bolyai Mathematical Society, pp. 535583.
[26] Sapozhenko, A. A. (1987) On the number of connected subsets with given cardinality of the boundary in bipartite graphs (in Russian). Metody Diskret. Analiz. 45 4270.
[27] Saxton, D. and Thomason, A. (2015) Hypergraph containers. Inventio Math. 201 925992.
[28] Schacht, M. (2016) Extremal results for random discrete structures. Ann. of Math. 184 333365.
[29] Sperner, E. (1928) Ein Satz über Untermengen einer endlichen Menge. Math. Z. 27 544548.
[30] Szemerédi, E. (1975) On sets of integers containing no k elements in arithmetic progression. Acta Arith. 27 199245.
[31] Turán, P. (1941) Eine Extremalaufgabe aus der Graphentheorie. Mat. Fiz Lapook 48 436452.

MSC classification

On Erdős–Ko–Rado for Random Hypergraphs II

  • A. HAMM (a1) and J. KAHN (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed