MathDB

In a tennis tournament where

n

players

A_1,A_2,\dots,A_n

take part, any two players play at most one match, and k \leq \frac {n(n \minus{} 1)}{2}

2

matches are played. The winner of a match gets

1

point while the loser gets

0

. Prove that a sequence

d_1,d_2,\dots,d_n

of nonnegative integers can be the sequence of scores of the players (

d_i

being the score of

A_i

) if and only if (i)\ \ d_1 \plus{} d_2 \plus{} \dots \plus{} d_n \equal{} k, and

(ii)\ \text{for any} X\subset\{A_1,\dots,A_n\}

, the number of matches between the players in

X

is at most

\sum_{A_j\in X}d_j

Problem Statement