# College math questions and answers

Recent questions in Post Secondary

2022-05-22

### Bounding the order of tournaments without transitive subtournaments of certain size.A tournament of order N is a directed graph on [N] obtained by assigning a direction to each edge of ${K}_{N}$. A tournament D is transitive if for every triple $a,b,c\in N$, $\left(a,b\right),\left(b,c\right)\in E\left(D\right)$ implies that $\left(a,c\right)\in E\left(D\right)$. For $n\in \mathbb{N}$ let f(n) be the maximum integer such that there exists a tournament of order f(n) without a transitive sub-tournament of size n. Show that $f\left(n\right)>\left(1+o\left(1\right)\right){2}^{\frac{n-1}{2}}$.

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