# Cycles of length three and four in tournaments

## TL;DR

This paper proves a conjecture about the minimal number of 4-cycles in large tournaments with many 3-cycles, showing that a specific random blow-up structure is asymptotically optimal for certain densities.

## Contribution

It confirms Linial and Morgenstern's conjecture for densities above 1/36 and characterizes the extremal tournament structures for densities above 1/16.

## Key findings

- Proves the conjecture for d ≥ 1/36.
- Characterizes extremal examples for d ≥ 1/16.
- Analyzes the spectrum of adjacency matrices of tournaments.

## Abstract

Linial and Morgenstern conjectured that, among all $n$-vertex tournaments with $d\binom{n}{3}$ cycles of length three, the number of cycles of length four is asymptotically minimized by a random blow-up of a transitive tournament with all but one part of equal size and one smaller part. We prove the conjecture for $d\ge 1/36$ by analyzing the possible spectrum of adjacency matrices of tournaments. We also demonstrate that the family of extremal examples is broader than expected and give its full description for $d\ge 1/16$.