A new generic class of Frankl’s families

Pierre Colomb, Alexis Irlande, Olivier Raynaud, Yoan Renaud


Frankl’s conjecture states that in a family of sets closed by union F such that F 6= {∅}, there is an element that belongs to at least half of the sets of F. There are several partial results of this conjecture. For example, it has been shown that families in which the smallest set is of size 1 or 2, or families closed both by union and by intersection are Frankl’s. In this article, by basing ourselves on an unseen recursive definition of the family of sets closed by union, we will define a new class of Frankl’s families. Subsequently, we will evaluate the size of this class for the first 6 values of n. Finally we will show that this class does not coincide with the already known Frankl’s classes.

Texto completo:



On Graphs, N.A.S.I., Order, Rival, I., Organization., N.A.T. Graphs and order: the role of graphs in the theory of ordered sets and its applications / edited by Ivan Rival. D. Reidel Pub. Co.; Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, Dordrecht, Holland; Boston: Hingham, MA, U.S.A.(1985)

D. G. Sarvate, J.C.R.: On the union-closed sets conjecture. Ars Combin. 27 149–153 (1989)

Frankl, P.: Extremal set systems. 1293–1329 (1995)

Stanley, R.P.: Enumerative Combinatorics, Vol I. The Wadsworth and Brooks Cole Mathematics Series (1986)

Abe, T.: Strong semimodular lattices and frankl’s conjecture. Algebra Universalis 44 (2000)

Poonen, B.: Union-closed families. J. Comb. Theory Ser. A 59(2) 253–268 (1992)

Abe, T., Nakano, B.: Frankl s conjecture is true for modular lattices. Graphs and Combinatorics 14 305–311 (1998)

Abe, T., Nakano, B.: Lower semimodular types of lattices: Frankl s conjecture holds for lower quasi-semimodular lattices. Graphs and Combinatorics 16 1–16 (2000)

Ivica Bosnjak, P.M.: The 11-element case of frankl’s conjecture. The electronic journal of combinatorics 15 (2008)

C’, P.M. An attempt at frankl s conjecture (2007)

Johnson, R.T., Vaughan, T.P.: On union-closed families, i. J. Comb. Theory Ser. A 84(2) 242–249 (1998)

Morris, R.: Fc-families and improved bounds for frankl s conjecture. Eur. J. Comb. 27(2) 269–282 (2006)

Vaughan, T.P.: Families implying the frankl conjecture. Eur. J. Comb. 23(7) 851–860 (2002)

Colomb, P., Irlande, A., Raynaud, O.: Counting of moore families on n = 7. In: ICFCA, LNAI 5986. (2010)

DOI: http://dx.doi.org/10.15765/e.v1i1.184

Enlaces refback

  • No hay ningún enlace refback.