## Combinatorics Seminar

When: Sunday, December 28, 10am
Where: Schreiber 309
Speaker: Dan Hefetz, University of Birmingham
Title: Universality of graphs with few triangles and
anti-triangles
## Abstract:

In this talk we will discuss *3-random-like
graphs*, that is,
sequences of graphs in which the densities of triangles and
anti-triangles converge to 1/8. Since the random graph G(n,1/2)
is, in particular, 3-random-like, this can be viewed as a weak
version of quasirandomness. We first show that 3-random-like graphs
are 4-universal, that is, they contain induced copies of all
4-vertex graphs. This settles a question of Linial and Morgenstern.
We then show that for larger subgraphs, 3-random-like sequences
demonstrate a completely different behaviour. We show that for
every graph H on n≥13 vertices there exist 3-random-like graphs
without an induced copy of H. Moreover, we show that for every
ℓ there are 3-random-like graphs which are ℓ-universal but not
m-universal when m is sufficiently large compared to ℓ.
Based on joint work with Mykhaylo Tyomkyn.