Graph Homomorphism Densities

Razborov's flag algebras — triangle density vs edge density inequalities

Host Graph

Subgraph H

Edge density t(K₂,G):
Triangle density t(K₃,G):
t(H,G):
Kruskal-Katona bound:
t(H,G) = fraction of injective maps V(H)→V(G) preserving edges.

Razborov 2007: t(K₃) ≥ t(K₂)³ (Kruskal-Katona for graphs).

The curve t(△) = t(e)^(3/2) is the Kruskal-Katona bound — tight for Turán graphs T(n,2).