We study forcing pairs for quasirandom graphs. Chung, Graham, and Wilson initiated the study of families
of graphs with the property that if a large graph
has approximately homomorphism density
for some fixed
is quasirandom with density
. Such families
are said to be forcing. Several forcing families were found over the last three decades and characterizing all bipartite graphs
is a forcing pair is a well-known open problem in the area of quasirandom graphs, which is closely related to Sidorenko’s conjecture. In fact, most of the known forcing families involve bipartite graphs only.
We consider forcing pairs containing the triangle
. In particular, we show that if
is a forcing pair, then so is
is obtained from
by replacing every edge of
by a triangle (each of which introduces a new vertex). For the proof we first show that
is a forcing pair, which strengthens related results of Simonovits and Sós and of Conlon et al.