On the Erdös-Ginzburg-Ziv theorem and the Ramsey numbers for stars and matchings

Abstract 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 ( K 2 m −1 )→{0, 1,…, m −1} is a mapping of the edges of the complete graph on 2 m −1 vertices into {0, 1,…, m −1}, then there exists a star K 1,m in K 2 m −1 with edges e 1 , e 2 ,…, e m such that c ( e 1 )+ c ( e 2 )+⋯+ c ( e m )≡0 (mod m ). Theorem 8. Let m be an integer. If c : e ( K r ( r +1) m −1 )→{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 Z 1 , Z 2 ,…, Z m of S such that c ( Z 1 )+ c ( Z 2 )+⋯+ c ( Z m )≡0 (mod m ).