## Abstract

Let G be a bipartite graph, with k|e(G). The zero-sum bipartite Ramsey number B(G, Z_{k}) is the smallest integer t such that in every Z_{k}-coloring of the edges of K_{t,t}, there is a zero-sum mod k copy of G in K_{t,t}. In this article we give the first proof that determines B(G, Z_{2}) for all possible bipartite graphs G. In fact, we prove a much more general result from which B(G, Z_{2}) can be deduced: Let G be a (not necessarily connected) bipartite graph, which can be embedded in K_{n,n}, and let F be a field. A function f : E(K_{n,n}) → F is called G-stable if every copy of G in K_{n,n} has the same weight (the weight of a copy is the sum of the values of f on its edges). The set of all G-stable functions, denoted by U(G, K_{n,n}, F) is a linear space, which is called the K_{n,n} uniformity space of G over F. We determine U(G, K_{n,n}, F) and its dimension, for all G, n and F. Utilizing this result in the case F = Z_{2}, we can compute B(G, Z_{2}), for all bipartite graphs G.

Original language | English |
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Pages (from-to) | 151-166 |

Number of pages | 16 |

Journal | Journal of Graph Theory |

Volume | 29 |

Issue number | 3 |

DOIs | |

State | Published - Nov 1998 |

## Keywords

- Bipartite graphs
- Ramsey numbers
- Vector space
- Zero-sum

## ASJC Scopus subject areas

- Geometry and Topology