---
author:
- 'Matija Bucić[^1]'
- 'Eoin Long[^2]'
- 'Asaf Shapira[^3]'
- 'Benny Sudakov[^4]'
title: Tournament quasirandomness from local counting
---

Introduction
============

A combinatorial structure is said to be ‘quasirandom’ if it behaves in a similar manner to a random structure, where the comparison is made with respect to some deterministic property. The systematic study of quasirandomness was initiated by Thomason [@Th1] [@Th2], and Chung, Graham and Wilson [@CGW], who examined notions of quasirandomness arising from various graph properties. One of the surprising conclusions of these papers is that a wide-range of natural graph properties all lead to essentially the same notion of quasirandomness, in the sense that a graph satisfying one of the properties necessarily satisfies them all. Since then, notions of quasirandomness have been extensively studied in a wide variety of contexts, including hypergraphs [@CGsetsystems], [@GowHR], [@NRS], [@RS], permutations [@Cooper], [@KP] and groups [@Gow]. The reader is referred to the survey [@KS] for an overview of this extensive topic.

In this paper we will study notions of quasirandomness for tournaments. The first paper on this topic is due to Chung and Graham [@CG] who proved that, as in the graph case, a wide range of natural tournament 