Subgraph densities in $K_r$-free graphs

TL;DR

This paper disproves a conjecture about subgraph counts in $K_r$-free graphs, proposes an alternative, and establishes tight bounds for bipartite subgraphs of radius two in triangle-free graphs.

Contribution

It challenges existing conjectures on subgraph densities in $K_r$-free graphs and provides new bounds for bipartite graphs in triangle-free graphs.

Findings

Disproved a conjecture on subgraph counts in $K_r$-free graphs

Proposed an alternative conjecture for subgraph densities

Established asymptotically tight bounds for bipartite graphs of radius at most 2 in triangle-free graphs

Abstract

In this paper we disprove a conjecture of Lidick\'y and Murphy about the number of copies of a given graph in a $K_r$-free graph and give an alternative general conjecture. We also prove an asymptotically tight bound on the number of copies of any bipartite graph of radius at most $2$ in a triangle-free graph.