Theorem (Erdos 1963) If ()(1-2-)<1 then there is a k-paradoxical tournament of n players. Pick a random tournament T on n players [n]. Event As no player in \S beat all players in S. PrAs=(（1-2k)”- Pr As≤∑(1-2)m-k<1 s∈() s∈(R) If ￾n k ⇥ ￾1 ￾ 2￾k⇥n￾k < 1 then there is a k-paradoxical tournament of n players. Theorem (Erdős 1963) Pick a random tournament T on n players [n]. Event AS : no player in V \S beat all players in S. Pr[AS] = ￾ 1 ￾ 2￾k⇥n￾k Pr < 1 ￾ ⇧ ⇤ ⌥ S￾( [n] k ) AS ⇥ ⌃ ⌅ ⇥ ￾ S⇥( [n] k ) (1 ￾ 2￾k) n￾k