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 K4 in place of K3, but this was disproved by Thomason. But an analogous statement does hold for some other graphs, which are called common graphs. A characterization of common graphs seems, however, out of reach.Franek and Rödl proved that 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, Štovíček and Thomason by showing that no graph containing 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.

Local aspects of the Sidorenko property for linear equations

“…This result is perhaps slightly less trivial than it sounds; for example in the graphtheoretic setting, it is not known whether local commonness is equivalent to commonness (see [CHL22]). One of the reasons we have chosen to study these local properties is that all existing proofs in the literature that systems are not Sidorenko [KLM21b, Ver21, SW17, FPZ21] prove the stronger statement that in fact these systems are not locally Sidorenko.…”