Locally common graphs

Goodman proved that the sum of the number of triangles in a graph on [Formula: see text] nodes and its complement is at least [Formula: see text]; 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 [Formula: see text] in place of [Formula: see text] , but this was disproved by Thomason. But an analogous statement does hold for some other graphs, which are called common graphs. Characterization of common graphs seems, however, out of reach. Franek and Rödl proved that [Formula: see text] 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, Štovíček and Thomason by showing that no graph containing [Formula: see text] 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.