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# Journal of Combinatorics

## Volume 4 (2013)

### Number 2

### On the triangle space of a random graph

Pages: 229 – 249

DOI: https://dx.doi.org/10.4310/JOC.2013.v4.n2.a4

#### Authors

#### Abstract

Settling a first case of a conjecture of M. Kahle on the homology of the clique complex of the random graph $G=G_{n,p}$, we show, roughly speaking, that (with high probability) the triangles of $G$ span its cycle space whenever each of its edges lies in a triangle (which happens (w.h.p.) when $p$ is at least about $\sqrt{(3/2)\ln n/n}$, and not below this unless $p$ is very small). We give two related proofs of this statement, together with a fundamental “stability” theorem for triangle-free subgraphs of $G_{n,p}$, originally due to Kohayakawa, Łuczak and Rödl, that underlies the first of our proofs.

#### Keywords

Kahle’s conjecture, homology of the clique complex, threshold, stability theorem

#### 2010 Mathematics Subject Classification

05C35, 05C80, 05D40, 55U10, 60C05

Published 13 August 2013