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Supersaturation of even linear cycles in linear hypergraphs

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  23 June 2020

Tao Jiang  and
Liana Yepremyan
Tao Jiang
Affiliation:
1Department of Mathematics, Miami University, Oxford, OH45056, USA
Liana Yepremyan*
Affiliation:
2Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL 60607, USA, and Department of Mathematics, London School of Economics, London WC2A 2AE, UK
*
*Corresponding author. Email: [email protected]
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Abstract

A classical result of Erdős and, independently, of Bondy and Simonovits [3] says that the maximum number of edges in an n-vertex graph not containing C2k, the cycle of length 2k, is O(n1+1/k). Simonovits established a corresponding supersaturation result for C2k’s, showing that there exist positive constants C,c depending only on k such that every n-vertex graph G with e(G)⩾ Cn1+1/k contains at least c(e(G)/v(G))2k copies of C2k, this number of copies tightly achieved by the random graph (up to a multiplicative constant).

In this paper we extend Simonovits' result to a supersaturation result of r-uniform linear cycles of even length in r-uniform linear hypergraphs. Our proof is self-contained and includes the r = 2 case. As an auxiliary tool, we develop a reduction lemma from general host graphs to almost-regular host graphs that can be used for other supersaturation problems, and may therefore be of independent interest.

MSC classification

Primary: 05C65: Hypergraphs
Secondary: 05D05: Extremal set theory
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 29 , Issue 5 , September 2020 , pp. 698 - 721
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

†

Research supported in part by National Science Foundation grant DMS-1400249 and DMS-1855542. Email: [email protected]

‡

Research supported by Marie Sklodowska Curie Global Fellowship 846304.

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