On the logarithimic calculus and Sidorenko's conjecture

TL;DR

This paper introduces a logarithmic calculus method for proving subgraph density inequalities, successfully verifying Sidorenko's conjecture for new graph families and providing simplified proofs for existing results.

Contribution

It develops a novel logarithmic calculus approach and applies it to verify Sidorenko's and the forcing conjecture for specific bipartite graphs, advancing understanding in graph theory.

Findings

Verified Sidorenko's conjecture for new graph families

Provided a short analytic proof for a result by Conlon, Fox, and Sudakov

Proved the forcing conjecture for bipartite graphs with a complete vertex

Abstract

We study a type of calculus for proving inequalities between subgraph densities which is based on Jensen's inequality for the logarithmic function. As a demonstration of the method we verify the conjecture of Erd\"os-Simonovits and Sidorenko for new families of graphs. In particular we give a short analytic proof for a result by Conlon, Fox and Sudakov. Using this, we prove the forcing conjecture for bipartite graphs in which one vertex is complete to the other side.