On Some Optimal Constructions of Chess Tournament Combinations

An optimal solution for the following “chess tournament” problem is given. Let n, r be positive integers such that r<n. Put N=2n, R=2r+1. Let X_N,R be the set of all ordered pairs (T, A) of matrices of degree N such that T=(t_ij) is symmetric, A=(a_ij) is skew-symmetric, t_ij ∈,{0, 1, 2,…, R), a_ij ∈{0,1,–1}. Moreover, suppose t_ii=a_ii=0 (1?i?N). t_ij = t_ik>0 implies j=k, t_ij=0 is equivalent to a_ij=0, and |a_i1|+|a_i2|+…+|a_iN|=R (1?i?N). Let p(T, A) be the number of i such that 1?i?N and a_i1 + a_i2 + … + a_iN >0. The main result of this note is to show that max p(T, A) for (T, A)∈X_{N, R} is equal to [n(2r+1)/(r+1)], and a pair (T₀, A₀) satisfying p(T₀, A₀)=[n(2r+1)/(r+1)] is also given.