FKMO 2018 Problem 3

For 31 years, n (>6) tennis players have records of wins. It turns out that for every two players, there is a third player who has won over them before. Prove that for every integer $k,l$ such that $2^{2^k+1}-1>n, 1<l<2k+1$ , there exist $l$ players ( $A_1, A_2, ... , A_l$ ) such that every player $A_{i+1}$ won over $A_i$ . ( $A_{l+1}$ is same as $A_1$ )