TY - JOUR

T1 - Supersaturated Sparse Graphs and Hypergraphs

AU - Ferber, Asaf

AU - McKinley, Gweneth

AU - Samotij, Wojciech

N1 - Publisher Copyright:
© 2018 The Author(s). Published by Oxford University Press. All rights reserved.

PY - 2020/1/20

Y1 - 2020/1/20

N2 - A central problem in extremal graph theory is to estimate, for a given graph H, the number of H-free graphs on a given set of n vertices. In the case when H is not bipartite, Erdos, Frankl, and Rödl proved that there are 2(1+o(1))ex(n, H) such graphs. In the bipartite case, however, bounds of the form 2O(ex(n, H)) have been proven only for relatively few special graphs H. As a 1st attempt at addressing this problem in full generality, we show that such a bound follows merely from a rather natural assumption on the growth rate of n → ex(n, H); an analogous statement remains true when H is a uniform hypergraph. Subsequently, we derive several new results, along with most previously known estimates, as simple corollaries of our theorem. At the heart of our proof lies a general supersaturation statement that extends the seminal work of Erdos and Simonovits. The bounds on the number of H-free hypergraphs are derived from it using the method of hypergraph containers.

AB - A central problem in extremal graph theory is to estimate, for a given graph H, the number of H-free graphs on a given set of n vertices. In the case when H is not bipartite, Erdos, Frankl, and Rödl proved that there are 2(1+o(1))ex(n, H) such graphs. In the bipartite case, however, bounds of the form 2O(ex(n, H)) have been proven only for relatively few special graphs H. As a 1st attempt at addressing this problem in full generality, we show that such a bound follows merely from a rather natural assumption on the growth rate of n → ex(n, H); an analogous statement remains true when H is a uniform hypergraph. Subsequently, we derive several new results, along with most previously known estimates, as simple corollaries of our theorem. At the heart of our proof lies a general supersaturation statement that extends the seminal work of Erdos and Simonovits. The bounds on the number of H-free hypergraphs are derived from it using the method of hypergraph containers.

UR - http://www.scopus.com/inward/record.url?scp=85081751164&partnerID=8YFLogxK

U2 - 10.1093/imrn/rny030

DO - 10.1093/imrn/rny030

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AN - SCOPUS:85081751164

SN - 1073-7928

VL - 2020

SP - 378

EP - 402

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 2

ER -