# Clique factors in pseudorandom graphs

### Patrick Morris

Department of Mathematics, Universitat Politècnica de Catalunya, Barcelona, Spain

## Abstract

An $n$-vertex graph is said to to be $(p,β)$*-bijumbled* if for any vertex sets $A,B⊆V(G)$, we have

We prove that for any $r∈N_{≥3}$ and $c>0$ there exists an $ε>0$ such that any $n$-vertex $(p,β)$-bijumbled graph with $n∈rN$, $p>0$, $δ(G)≥cpn$ and $β≤εp_{r−1}n$ contains a $K_{r}$-factor. This implies a corresponding result for the stronger pseudorandom notion of $(n,d,λ)$-graphs. For the case of triangle factors, that is, when $r=3$, this result resolves a conjecture of Krivelevich, Sudakov and Szabó from 2004 and it is tight due to a pseudorandom triangle-free construction of Alon. In fact, in this case even more is true: as a corollary to this result and a result of Han, Kohayakawa, Person and the author, we can conclude that the same condition of $β=o(p_{2}n)$ actually guarantees that a $(p,β)$-bijumbled graph $G$ contains every graph on $n$ vertices with maximum degree at most 2.

## Cite this article

Patrick Morris, Clique factors in pseudorandom graphs. J. Eur. Math. Soc. (2023), published online first

DOI 10.4171/JEMS/1388