On the graph limit question of Vera T. S\'os

TL;DR

This paper investigates whether edge counts in subgraphs can characterize dense graph limits, confirming this equivalence for quasirandom graphs and extending the result to a broader context.

Contribution

It proves that edge count distributions suffice to define dense graph limits on quasirandom graphs and generalizes this finding.

Findings

Edge counts characterize dense graph limits on quasirandom graphs.

The equivalence extends beyond quasirandom graphs.

Provides a generalized framework for graph limit characterization.

Abstract

In the dense graph limit theory, the topology of the set of graphs is defined by the distribution of the subgraphs spanned by finite number of random vertices. Vera T. S\'os proposed a question that if we consider only the number of edges in the spanned subgraphs, then whether it provides an equivalent definition. We show that the answer is positive on quasirandom graphs, and we prove a generalization of the statement.