$d^-_T(u) = |\{v\in V(T): \overrightarrow{vu} \in E(T)\}|$. A tournament $T$ is regular if $d^+_T(u) = d^-_T(u)$ for all $u\in V(T)$. For $U,W \subseteq V(T)$ we denote by $e(U,W)$ the number of edges starting in $U$ and ending in $W$.

We say that $T$ is an $n$-vertex tournament if $|V(T)| = n$. An ordering of $T$ is a bijective map $\sigma : V \to [n]$ and the set of all orderings of $T$ is naturally identified with $S_n$, the symmetric group on $n$ elements. An edge $\overrightarrow {uv} \in E$ is a $\sigma $-forward edge if $\sigma (u) < \sigma (v)$ and we write $F_{\sigma ,T} \subset E$ to denote the set of $\sigma $-forward edges of $T$. Let $B_{\sigma , T} = E \setminus F_{\sigma ,T}$, the set of $\sigma $-backward edges. $T$ is said to be transitive if $F_{\sigma ,T} = E$ for some $\sigma \in S_n$ and write $Tr_n$ to denote the unique (up to isomorphism) $n$-vertex transitive tournament. Lastly, we will write $a \pm b$ to denote some value $c$ with $a-b \leq c \leq a+b$.

Our starting point in this paper is a result due to Chung and Graham [@CG], which gives two equivalences of tournament 