Some two color, four variable Rado numbers

Abstract

There exists a minimum integer N such that any 2-coloring of {1, 2, . . . , N } admits a monochromatic solution to x + y + kz = ℓw for k, ℓ ∈ Z + , where N depends on k and ℓ. We determine N when ℓ − k ∈ {0, 1, 2, 3, 4, 5}, for all k, ℓ for which 1 2 ((ℓ − k) 2 − 2)(ℓ − k + 1) ≤ k ≤ ℓ − 4, as well as for arbitrary k when ℓ = 2.

New Upper and Lower Bounds on the Rado Numbers

“…Tables of computed 2 and 3-color Rado numbers are can be found in the appendix. We include an extension of the table of computed 2-color Rado numbers presented in Meyers and Robertson's paper [5], as well as the first published results on 3-color Rado numbers.…”

“…Upper Bound. Assume, for contradiction, that there exists a coloring of [1,5] Lower Bound. It is easy to see that the coloring RBRBBRBRBRRBRB admits no monochromatic solutions to 3x + 3y + az = (a + 5)w in [1,14].…”

“…Previous results have determined the Rado numbers for some classes of equations, completely characterized the 2-color Rado numbers for equations of the form a(x + y) = bz [7], while Robertson and Myers gave results and conjectures for four variable equations of the form of x + y + kz = jw [5]. Here we examine the case a(x − y) = bz and x + ay = abz, bounding the two color Rado numbers of both.…”

Bootstrapping partition regularity of linear systems

“…When A = (a 1 , a 2 , −a 2 ) for a 1 , a 2 ∈ N the value of R A (2) is computed in [26,Theorem 9.17]; when A = (a 1 , a 2 , −(a 1 + a 2 )) for a 1 , a 2 ∈ N the value of R A (2) is computed in [23, Theorem 1.1]; when A = (1, 1, a 3 , −a 4 ) for a 3 , a 4 ∈ N (where partition regularity of A ensures that a 4 ∈ {1, 2, a 3 , a 3 + 1, a 3 + 2}) the value of R A (2) is computed in [33,Theorems 3,4 and 8] for a 4 = a 3 , a 4 = a 3 + 1 and a 4 = 2 respectively, with the case a 4 = a 3 + 2 being trivial; and when…”

“…, a k ∈ N and n a n+1 + • • • + a k , the value of R A (2) is computed in [35,Theorem 3]. Note that the work of [26], [33] and [35] goes further and computes R A (2) for some A which are not partition regular. (This makes sense since we may have…”