A linear algebraic view of partition regular matrices

Abstract

1Rado showed that a rational matrix is partition regular over N if and only if it satisfies the columns 2 condition. We investigate linear algebraic properties of the columns condition, especially for oriented 3 (vertex-arc) incidence matrices of directed graphs and for sign pattern matrices. It is established that 4 the oriented incidence matrix of a directed graph Γ has the columns condition if and only if Γ is strongly 5 connected, and in this case an algorithm is presented to find a partition of the columns of the oriented 6 incidence matrix with the maximum number of cells. It is shown that a sign pattern matrix allows the 7 columns condition if and only if each row is either all zeros or the row has both a + and −. 8 AMS subject classifications: (2010) 15A03, 05C50, 15B35, 05D10.9

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“…The study of partition regularity has long been a combinatorial endeavor, which mostly uses the columns condition to check if a given matrix is partition regular. However, Hogben and McLeod [2010] recently showed that the columns condition is interesting in its own right, and provided a more linear algebraic perspective on partition regularity. We employ this new perspective to extend the notion of partition regularity into geometrical and topological settings.…”

“…and the minimum rank of G is [Barioli et al 2010;Edholm et al 2010;Hogben 2010;Huang et al 2010]. Let G be a graph with each vertex colored either white or black.…”

A rainbow Ramsey analogue of Rado’s theorem

“…For a given oriented graph G = (V, E) with |V | = n and |E| = m we consider the incidence matrix M which is a n × m matrix. The rank of M is n − c where c is the number of connected components of G. Theorem 1.5 has the following graph theoretic corollary (for a monochromatic analogue see [6]). In the third section we look at the matrices which give rise to Fibonacci sequences; we use Theorem 1.5 to show that they are rainbow regular and give a bound for their rainbow number.…”

“…For a given oriented graph G = (V, E) with |V | = n and |E| = m we consider the incidence matrix M which is a n × m matrix. The rank of M is n − c where c is the number of connected components of G. Theorem 1.5 has the following graph theoretic corollary (for a monochromatic analogue see [6]).…”

A topological generalization of partition regularity

“…The study of partition regularity has long been a combinatorial endeavor, which mostly uses the columns condition to check if a given matrix is partition regular. However, Hogben and McLeod [2010] recently showed that the columns condition is interesting in its own right, and provided a more linear 422 LIAM SOLUS algebraic perspective on partition regularity. We employ this new perspective to extend the notion of partition regularity into geometrical and topological settings.…”