On Rado's Boundedness Conjecture

Abstract

We prove that Rado's Boundedness Conjecture from Richard Rado's 1933 famous dissertation Studien zur Kombinatorik is true if it is true for homogeneous equations. We then prove the first nontrivial case of Rado's Boundedness Conjecture: if a 1 , a 2 , and a 3 are integers, and if for every 24-coloring of the positive integers (or even the nonzero rational numbers) there is a monochromatic solution to the equation a 1 x 1 + a 2 x 2 + a 3 x 3 = 0, then for every finite coloring of the positive integers there is a monochromatic solution to a 1 x 1 + a 2 x 2 + a 3 x 3 = 0.

On minimal colorings without monochromatic solutions to a linear equation

“…The following straightforward lemma [FK05] is useful in proving that certain colorings are free of monochromatic solutions to particular linear equations.…”

Recent progress in Ramsey theory on the integers

“…In fact, Fox and Radoicic show that dor(a, b) ≤ 6 for all (a, b) = (1, 1). Their proof makes use of a result, due to Fox and Kleitman [13], which states that if a linear homogeneous equation in three variables is 24-regular, then it is regular. In [15], it is shown that 2 ≤ dor(a, 2a − 2) ≤ 4 for all a ≥ 2; that dor(a, 2a + j) ≤ 4 for 1 ≤ j ≤ 5; and that dor(a, 2a + 1) ≤ 3 for all a ≥ 1.…”

“…For b ≥ 0, let g(b) be the least positive integer (if it exists) such that for every 2-coloring of [1, g(b)] there is a monochromatic set of the form x, x + d, x + 2d + b. They showed that for b even, 2b + 10 ≤ g(b) ≤ 13 2 b + 1. The upper bound was improved by the present paper's second author, to 9 4 b + 9, who also conjectured that for b ≥ 10 even, g(b) = 2b + 10 [20].…”

Maximum $k$-Sum $\mathbf{n}$-Free Sets of the 2-Dimensional Integer Lattice

“…The case when b = 0, which is also referred to as "L is homogeneous", was actively studied due to its close ties to other subjects such as Sidon sets, progression-free sets, and Rado's boundedness conjecture. See [12][13][14] for recent results on L-free sets where L is a homogeneous linear equation, and see [9,17] for details regarding Rado's boundedness conjecture. Also, the complexity of finding a maximum L-free set is known to be NP-complete in almost all cases, see [7,16] for recent results.…”