On partition and almost disjoint properties of combinatorial notions
Teng Zhang
TL;DR
The paper addresses infinite partition and almost disjoint properties for diverse notions of largeness in semigroups, extending beyond central sets to IP sets, combinatorially rich sets, $C_p$-sets, and PP-rich sets. It develops uncountable generalizations by leveraging Stone–Čech compactification techniques and polynomial extensions, establishing that IP and combinatorially rich notions exhibit strong partition and disjointness properties in many countable contexts, and proving an uncountable version of the polynomial extension of the central sets theorem for commutative cancellative semigroups. Key results include that IP sets in infinite left weakly cancellative semigroups split into $\omega$ disjoint IP subsets and host $2^\omega$ almost disjoint IP subsets, along with analogous findings for combinatorially rich and $C_p$-sets, and a uncountable polynomial-extended framework that subsumes the central set theorem. The work also highlights open questions for uncountable semigroups, particularly regarding $J_p$-sets and the full partition regularity of PP-rich and related notions.
Abstract
It is known that there are many notions of largeness in a semigroup that own rich combinatorial properties. In this paper, we focus on partition and almost disjoint properties of these notions. One of the most remarkable results with respect to this topic is that in an infinite very weakly cancellative semigroup of size κ, every central set can be split into κdisjoint central subsets. Moreover, if κcontains λalmost disjoint subsets, then every central set contains a family of λalmost disjoint central subsets. And many other combinatorial notions are found successively to have analogous properties, among these are thick sets, piecewise syndetic sets, J-sets and C-sets. In this paper, we mainly study four other notions: IP sets, combinatorially rich sets, Cp-sets and PP-rich sets. Where the latter two are known in (N, +), related to the polynomial extension of the central sets theorem. We lift them up to commutative cancellative semigroups and obtain an uncountable version of the polynomial extension of the central sets theorem incidentally. And we finally find that the infinite partition and almost disjoint properties hold for Cp-sets in commutative cancellative semigroups and for other three notions in (N, +).