AbstractGiven a fixed graph
$H$
and a constant
$c \in [0,1]$
, we can ask what graphs
$G$
with edge density
$c$
asymptotically maximise the homomorphism density of
$H$
in
$G$
. For all
$H$
for which this problem has been solved, the maximum is always asymptotically attained on one of two kinds of graphs: the quasi-star or the quasi-clique. We show that for any
$H$
the maximising
$G$
is asymptotically a threshold graph, while the quasi-clique and the quasi-star are the simplest threshold graphs, having only two parts. This result gives us a unified framework to derive a number of results on graph homomorphism maximisation, some of which were also found quite recently and independently using several different approaches. We show that there exist graphs
$H$
and densities
$c$
such that the optimising graph
$G$
is neither the quasi-star nor the quasi-clique (Day and Sarkar, SIAM J. Discrete Math. 35(1), 294–306, 2021). We also show that for
$c$
large enough all graphs
$H$
maximise on the quasi-clique (Gerbner et al., J. Graph Theory 96(1), 34–43, 2021), and for any
$c \in [0,1]$
the density of
$K_{1,2}$
is always maximised on either the quasi-star or the quasi-clique (Ahlswede and Katona, Acta Math. Hung. 32(1–2), 97–120, 1978). Finally, we extend our results to uniform hypergraphs.