Skip to main content
×
×
Home

On Colourings of Hypergraphs Without Monochromatic Fano Planes

  • HANNO LEFMANN (a1), YURY PERSON (a2), VOJTĚCH RÖDL (a3) and MATHIAS SCHACHT (a2)
Abstract

For k-uniform hypergraphs F and H and an integer r, let cr,F(H) denote the number of r-colourings of the set of hyperedges of H with no monochromatic copy of F, and let , where the maximum runs over all k-uniform hypergraphs on n vertices. Moreover, let ex(n,F) be the usual extremal or Turán function, i.e., the maximum number of hyperedges of an n-vertex k-uniform hypergraph which contains no copy of F.

For complete graphs F = K and r = 2, Erdős and Rothschild conjectured that c2,K(n) = 2ex(n,K). This conjecture was proved by Yuster for ℓ = 3 and by Alon, Balogh, Keevash and Sudakov for arbitrary ℓ. In this paper, we consider the question for hypergraphs and show that, in the special case when F is the Fano plane and r = 2 or 3, then cr,F(n) = rex(n,F), while cr,F(n) ≫ rex(n,F) for r ≥ 4.

Copyright
References
Hide All
[1]Alon, N., Balogh, J., Keevash, P. and Sudakov, B. (2004) The number of edge colorings with no monochromatic cliques. J. London Math. Soc. (2) 70 273288.
[2]Balogh, J. (2006) A remark on the number of edge colorings of graphs. Europ. J. Combin. 27 565573.
[3]Balogh, J., Bollobás, B. and Simonovits, M. (2004) The number of graphs without forbidden subgraphs. J. Combin. Theory Ser. B 91 124.
[4]Balogh, J., Bollobás, B. and Simonovits, M. (2009) The typical structure of graphs without given excluded subgraphs. Random Struct. Alg. 34 305318.
[5]Chung, F. R. K. (1991) Regularity lemmas for hypergraphs and quasi-randomness. Random Struct. Alg. 2 241252.
[6]Erdős, P. (1974) Some new applications of probability methods to combinatorial analysis and graph theory. In Proc. Fifth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic University, Boca Raton 1974), pp. 3951.
[7]Erdős, P., Frankl, P. and Rödl, V. (1986) The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent. Graphs Combin. 2 113121.
[8]Erdős, P., Kleitman, D. J. and Rothschild, B. L. (1976) Asymptotic enumeration of Kn-free graphs. In Colloquio Internazionale sulle Teorie Combinatorie (Rome 1973), Vol. II, Accad. Naz. Lincei, Rome, pp. 1927.
[9]Frankl, P. and Rödl, V. (1992) The uniformity lemma for hypergraphs. Graphs Combin. 8 309312.
[10]Füredi, Z. and Simonovits, M. (2005) Triple systems not containing a Fano configuration. Combin. Probab. Comput. 14 467484.
[11]Keevash, P. and Sudakov, B. (2005) The Turán number of the Fano plane. Combinatorica 25 561574.
[12]Kohayakawa, Y., Nagle, B., Rödl, V. and Schacht, M. Weak hypergraph regularity and linear hypergraphs. J. Combin. Theory, Sec. B, to appear.
[13]Kolaitis, P. G., Prömel, H. J. and Rothschild, B. L. (1985) Asymptotic enumeration and a 0–1 law for m-clique free graphs. Bull. Amer. Math. Soc. (N.S.) 13 160162.
[14]Kolaitis, P. G., Prömel, H. J. and Rothschild, B. L. (1987) K l+1-free graphs: Asymptotic structure and a 0–1 law. Trans. Amer. Math. Soc. 303 637671.
[15]Nagle, B. and Rödl, V. (2001) The asymptotic number of triple systems not containing a fixed one. Discrete Math. 235 271290.
[16]Nagle, B., Rödl, V. and Schacht, M. (2006) Extremal hypergraph problems and the regularity method. In Topics in Discrete Mathematics, Algorithms Combin., Vol. 26, Springer, Berlin, pp. 247278.
[17]Person, Y. and Schacht, M. (2009) Almost all hypergraphs without Fano planes are bipartite. In Proc. Twentieth Annual ACM–SIAM Symposium on Discrete Algorithms (Mathieu, C., ed.), ACM, pp. 217226.
[18]Steger, A. (1990) Die Kleitman–Rothschild Methode. PhD thesis, Forschungsinstitut für Diskrete Mathematik, Rheinische Friedrichs–Wilhelms–Universität Bonn.
[19]Szemerédi, E. (1978) Regular partitions of graphs. In Problèmes Combinatoires et Théorie des Graphes (Colloq. Internat. CNRS, Orsay 1976), CNRS, Paris, pp. 399401.
[20]Yuster, R. (1996) The number of edge colorings with no monochromatic triangle. J. Graph Theory 21 441452.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

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