# More non-bipartite forcing pairs

## TL;DR

This paper investigates pairs of graphs called forcing pairs, focusing on non-bipartite cases, and establishes that a specific number of doublings always suffices to ensure quasirandomness.

## Contribution

It proves that for any t>3, the minimum number of doublings needed for forcing pairs is always (t+2)/2, extending previous results.

## Key findings

- (K_t,F) is forcing where F is obtained by iterative doubling of K_t
- Two doublings suffice for t=3 to establish forcing pairs
- For all t>3, (t+2)/2 doublings are enough

## Abstract

We study pairs of graphs (H_1,H_2) such that every graph with the densities of H_1 and H_2 close to the densities of H_1 and H_2 in a random graph is quasirandom; such pairs (H_1,H_2) are called forcing. Non-bipartite forcing pairs were first discovered by Conlon, Han, Person and Schacht [Weak quasi-randomness for uniform hypergraphs, Random Structures Algorithms 40 (2012), 1-38]: they showed that (K_t,F) is forcing where F is the graph that arises from K_t by iteratively doubling its vertices and edges in a prescribed way t times. Reiher and Schacht [Forcing quasirandomness with triangles, Forum of Mathematics, Sigma 7, 2019] strengthened this result for t=3 by proving that two doublings suffice and asked for the minimum number of doublings needed for t>3. We show that (t+2)/2 doublings always suffice.