Hereditary quasirandom properties of hypergraphs
- 77 Downloads
Thomason and Chung, Graham and Wilson were the first to systematically investigate properties of quasirandom graphs. They have stated several quite disparate graph properties — such as having uniform edge distribution or containing a prescribed number of certain subgraphs — and proved that these properties are equivalent in a deterministic sense.
Simonovits and Sós introduced a hereditary property (which we call S) stating the following: for a small fixed graph L, a graph G on n vertices is said to have the property S if for every set X ⊆ V(G), the number of labeled copies of L in G[X] (the subgraph of G induced by the vertices of X) is given by 2−e(L)|X| υ(L) + o(n υ(L)). They have shown that S is equivalent to the other quasirandom properties.
In this paper we give a natural extension of the result of Simonovits and Sós to k-uniform hypergraphs, answering a question of Conlon et al. Our approach also yields an alternative, and perhaps simpler, proof of one of their theorems.
Mathematics Subject Classification (2000)05C65
- David Conlon, Hiêp Hàn, Yury Person and Mathias Schacht: Weak quasirandomness for uniform hypergraphs, (2009), to appear in Random Structures and Algorithms.Google Scholar
- Endre Szemerédi: Regular partitions of graphs, in: Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), volume 260 of Colloq. Internat. CNRS, pages 399–401, CNRS, Paris, 1978.Google Scholar
- Andrew Thomason: Pseudorandom graphs, in: Random graphs’ 85 (Poznań, 1985), volume 144 of North-Holland Math. Stud., pages 307–331, North-Holland, Amsterdam, 1987.Google Scholar