Hereditary quasirandom properties of hypergraphs

Abstract

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 XV(G), the number of labeled copies of L in G[X] (the subgraph of G induced by the vertices of X) is given by 2e(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.

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Correspondence to Domingos Dellamonica Jr..

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Supported by a CAPES/Fulbright scholarship.

Partially supported by NSF grant DMS0800070.

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Dellamonica, D., Rödl, V. Hereditary quasirandom properties of hypergraphs. Combinatorica 31, 165 (2011). https://doi.org/10.1007/s00493-011-2621-8

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Mathematics Subject Classification (2000)

  • 05C65