DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory 24(3) (2004) 469-484
DOI: 10.7151/dmgt.1245

SHORT PATHS IN 3-UNIFORM QUASI-RANDOM HYPERGRAPHS

Joanna Polcyn

Department of Discrete Mathematics
Adam Mickiewicz University
Poznań

e-mail: joaska@amu.edu.pl

Abstract

Frankl and Rödl [3] proved a strong regularity lemma for 3-uniform hypergraphs, based on the concept of δ-regularity with respect to an underlying 3-partite graph. In applications of that lemma it is often important to be able to ``glue" together separate pieces of the desired subhypergraph. With this goal in mind, in this paper it is proved that every pair of typical edges of the underlying graph can be connected by a hyperpath of length at most seven. The typicality of edges is defined in terms of graph and hypergraph neighborhoods, and it is shown that all but a small fraction of edges are indeed typical.

Keywords: hypergraph, path, quasi-randomness.

2000 Mathematics Subject Classification: 05C65, 05C12, 05C38.

References

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[7] J. Polcyn, V. Rödl, A. Ruciński and E. Szemerédi, Short paths in quasi-random triple systems with sparse underlying graphs, in preparation.
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[9] V. Rödl, A. Ruciński and E. Szemerédi, A Dirac-type theorem for 3-uniform hypergraphs, submitted.
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Received 18 June 2003
Revised 21 January 2004