New polynomial and multidimensional extensions of classical partition\n results

2015/01/10 by Vitaly Bergelson, Bergelson, Vitaly, John H. Johnson +3
Computer Science · Mathematics · #Advanced Algebra and Logic #Combinatorics (math.CO) #FOS: Mathematics #Functional Equations Stability Results #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1501.02408

openalex publication_date 2015/01/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In the 1970s Deuber introduced the notion of (m,p,c)-sets in \ℕ\nand showed that these sets are partition regular and contain all linear\npartition regular configurations in \ℕ. In this paper we obtain\nenhancements and extensions of classical results on (m,p,c)-sets in two\ndirections. First, we show, with the help of ultrafilter techniques, that\nDeuber's results extend to polynomial configurations in abelian groups. In\nparticular, we obtain new partition regular polynomial configurations in\n\ℤ^d. Second, we give two proofs of a generalization of Deuber's\nresults to general commutative semigroups. We also obtain a polynomial version\nof the central sets theorem of Furstenberg, extend the theory of\n(m,p,c)-systems of Deuber, Hindman and Lefmann and generalize a classical\ntheorem of Rado regarding partition regularity of linear systems of equations\nover \ℕ to commutative semigroups.\n

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