Question 154
Let $N \gt M \gt 1$ be fixed integers. There are $N$ people playing in a chess tournament; each pair of players plays each other once, with no draws. It turns out that for each sequence of $M + 1$ distinct players $P_0, P_1, \dots, P_M$ such that $P_{i-1}$ beat $P_i$ for each $i = 1, \dots, M$, player $P_0$ also beat $P_M$. Prove that the players can be numbered $1, 2, \dots, N$ in such a way that, whenever $a \ge b + M - 1$, player $a$ beat player $b$.