Affine and combinatorial binary m-spaces (Q1063235)

Let V(n) denote the n-dimensional vector space over GF(2). Let a(m,r) (respectively, c(m,r)) denote the smallest positive integer such that if \(n\geq a(m,r)\) (respectively, \(n\geq c(m,r))\), and V(n) is arbitrarily partitioned into r classes \(C_ i\), \(1\leq i\leq r\), then some class \(C_ i\) must contain an m-dimensional affine (respectively, combinatorial) subspace of V(n). The author investigates upper bounds for the functions a(m,r) and c(m,r) and for the corresponding ''density functions'' \(\bar a(\)m,\(\epsilon)\) and \(\bar c(\)m,\(\epsilon)\).