ON THE ERDOS-GINZBURG-ZIV THEOREM AND THE RAMSEY NUMBERS FOR STARS AND MATCHINGS

A link between Ramsey numbers for stars and matchings and the Erdos-Ginzburg-Ziv theorem is established. Known results are generalized. Among other results we prove the following two theorems. Theorem 5. Let m be an even integer. If c : e (K2m-1) --> {0,1,...,m-1} is a mapping of the edges of the complete graph on 2m - 1 vertices into {0,1,...,m-1}, then there exists a star K1,m in K2m-1 with edges e1,e2,...,e(m) such that c(e1) + c(e2) +...+ c(e(m)) = 0 (mod m). Theorem 8. Let m be an integer. If c : (K(r+1)m-1)r) --> {0,1...,m-1} is a mapping of all the r-subsets of an (r + 1)m - 1 element set S into {0,1,..., m-1}, then there are m pairwise disjoint r-subsets Z1,Z2,...,Z(m) of S such that c(Z1) + c(Z2) +...+ c(Z(m)) = 0 (mod m).