The dimension of the region of feasible tournament profiles

TL;DR

This paper determines the dimension of the feasible profile region for k-vertex tournaments, revealing it is significantly larger than the count of strongly connected tournaments, with connections to Lyndon words.

Contribution

It extends the understanding of feasible profile regions from graphs to tournaments, establishing the dimension and uncovering a link to Lyndon words.

Findings

Dimension of tournament profile region is much larger than the number of strongly connected tournaments.

The result reveals a surprising connection to Lyndon words.

Provides a new perspective on the structure of tournament profiles.

Abstract

Erd\H os, Lov\'asz and Spencer showed in the late 1970s that the dimension of the region of $k$-vertex graph profiles, i.e., the region of feasible densities of $k$-vertex graphs in large graphs, is equal to the number of non-trivial connected graphs with at most $k$ vertices. We determine the dimension of the region of $k$-vertex tournament profiles. Our result, which explores an interesting connection to Lyndon words, yields that the dimension is much larger than just the number of strongly connected tournaments, which would be the answer expected as the analogy to the setting of graphs.