Locally common graphs

Goodman proved that the sum of the number of triangles in a graph on n n nodes and its complement is at least n3/24 n3\unicode{x02215}24; in other words, this sum is minimized, asymptotically, by a random graph with edge density 1/2. Erdos conjectured that a similar inequality will hold for K4 K4 in place of K3 K3, 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 Rodl proved that K4 K4 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, Stovicek and Thomason by showing that no graph containing K4 K4 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.