Locally common graphs

Abstract

Goodman proved that the sum of the number of triangles in a graph on $n$ nodes and its complement is at least $n^3/24$; in other words, this sum is minimized, asymptotically, by a random graph with edge density $1/2$. Erdős conjectured that a similar inequality will hold for $K_4$ in place of $K_3$, but this was disproved by Thomason. But an analogous statement does hold for some other graphs, which are called {\it common graphs}. A characterization of common graphs seems, however, out of reach. Franek and Rödl proved that $K_4$ is common in a weaker, local sense. Using the language of graph limits, we study two versions of locally common graphs. We sharpen a result of Jagger, Štov\'ıček and Thomason by showing that no graph containing $K_4$ can be locally common, but prove that all such graphs are weakly locally common. We also show that not all connected graphs are weakly locally common.