On Ramsey Properties of k-Majority Tournaments

A central objective in Ramsey theory is determining whether restricted families of discrete structures necessarily contain substantially larger homogeneous substructures, compared to the unrestricted structures. In the setting of tournaments, it is well known that every tournament contains a transitive subgraph of size $\log n$, and that this is best possible up to a constant factor. A restricted family of tournaments that has been extensively studied is the family of $k$-majority tournaments. They are obtained by taking $2k-1$ linear orders of a set $X$, and defining a tournament on $X$ which has an edge from $u$ to $v$ if $u$ precedes $v$ in at least $k$ of these orders. Milans, Schreiber, and West proved that such tournaments indeed have significantly larger transitive tournaments. More precisely, they proved that every $k$-majority tournament contains a transitive tournament of size $n^{2^{-\Theta(k)}}$. Our main goal in this paper is to give an exponential improvement in the dependence of the exponent on $k$ by showing that every $k$-majority tournament contains a transitive set of size $n^{\Omega(1/k)}$. Finally, we highlight several open problems and conjectural directions related to random $k$-majority tournaments.