Partition Theorems for Subsets of Vector Spaces

1. INTRODUCTION II [1] two of us iivestigated the problem of determiiiig those cardiials a, /3, y, 8, A for which the followiig statemeet, abbreviated Iá(8, g, a, A, y), holds : "Wheeever V is aa a-dimeesiooal vector space over a field of A elemeets, aad the 8-dimeesiooal subspaces of V are partitiooed iito y classes, there is some /-dimeesiooal subspace of V all of whose 8-dimee-siooal subspaces are ii the same class ." II this paper, we iivestigate the related questioo of which cardiials a, /3, y, aad 8 make the followiig statemeet valid : "Wheeever V is aa a-dimeesiooal vector space over GF(2) aad V = Ua (We could of course ask the same questioo with a field of A elemeets replaciig GF(2). However, we have o iiterestiig results whee A :A 2 .) Note that the statemeet -> ooly makes seese if 8 < w. The statemeet has a simple set-theoretic formulatioo ii terms of the symmetric differeece, d, of two sets. We will use the otatioo d~_~A j = (J i=,A i) JA t. We also take a cardiial to be the least ordiial