Abstract
We fully characterise the solvability of Rado equations inside linear combinations a1U+…+anU of idempotent ultrafilters U∈βZ by exploiting known relations between such combinations and strings of integers. This generalises a partial characterisation obtained previously by Mauro Di Nasso.
Highlights
A long studied problem in combinatorics deals with the partition regularity of Diophantine equations
We fully characterise the solvability of Rado equations inside linear combinations a1U + · · · + anU of idempotent ultrafilters U ∈ βZ by exploiting known relations between such combinations and strings of integers
The above result is the major inspiration for this work; in this paper, we extend Di Nasso’s result by finding a complete characterisation of which Rado equations are solved by linear combinations of the form a1U + · · · + anU for a1, . . . , an ∈ Z and U ∈ βZ an additively idempotent ultrafilter; dropping Di Nasso’s assumptions that c1 +· · ·+cm = 0 and U ∈ βN
Summary
A long studied problem in combinatorics deals with the partition regularity of Diophantine equations. Ultrafilters and their algebra have been useful in the study of partition regularity of equations and systems of equations since Galvin and Glazer’s proof of Hindman Theorem; for classical results such as. The proof of (iv) is a consequence of Theorem 2 of [1], in which the authors proved that any additively minimal idempotent element of βS witnesses the partition regularity of any Rado equation. We will use the above results only in the linear case, we recall here that Theorem 2.5 displayed a major importance in the study of the partition regularity of nonlinear polynomials in [4, 5, 13, 14]
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