Intersections of Oriented Graphs and Tournaments
Given two tournaments of order $n$, Bollob\'as and Scott defined their discrepancy as the largest deviation of their overlap from its random average under relabelling, and they asked whether the resulting discrepancy is always $\Omega (n^{3/2})$. We answer this question by proving that there is an absolute constant $c>0$ such that every pair of tournaments $T,U$ of order $n$ has discrepancy at least $cn^{3/2}$. More generally, if $D$ and $H$ are oriented graphs of order \( n \) with \( e(D) = p \binom{n}{2} \) and \( e(H) = q \binom{n}{2} \) satisfying $16/n \leq p, q \leq 1 - 16/n$, then there is an absolute constant \( c > 0 \) such that their discrepancy is at least $c(p(1-p)q(1-q))^{3}n^{3/2}$. We also show that these two-graph estimates extend to intersections of any fixed number of graphs, tournaments, and oriented graphs.