Some Partition Theorems for Infinite and Finite Matrices

Let A be a finite or infinite matrix with integer entries and only finitely many nonzero entries in each row. Then A is image partition regular (over N) provided whenever N is finitely colored, there must exist ~x with entries from N (and the same number of entries as A has columns) such that the entries of A~x are the same color. Let k 2 N and let ~a = ha1, a2, . . . , aki be a sequence in Z \ {0}. Let B(~a) denote an infinite matrix consisting of all rows whose nonzero entries are a1, a2, . . . , ak in order, each occurring once, and for m k, Bm(~a) is a matrix with m columns and all rows whose nonzero entries are a1, a2, . . . , ak in order, each occurring once. Also define M(~a) = ✓ I B(~a) ◆ and Mm(~a) = ✓ Im Bm(~a) ◆ , where I is the !⇥! identity matrix and Im is the m⇥m identity matrix. We provide a simple characterization of those sequences ~a with the property that for su ciently large m, Mm(~a) is image partition regular. We also provide a simple characterization of those sequences ~a such that M(~a) is image partition regular in the special case where there is a fixed 2 N such that each ai is either a power of or the negative of a power of . 1This author acknowledges support from NSERC Discovery grant 228064. 2This author acknowledges support received from the National Science Foundation (USA) under grant DMS-1160566. INTEGERS: 14 (2014) 2