Some new results in multiplicative and additive Ramsey theory

Abstract:

There are several notions of largeness that make sense in any semigroup, and others such as the various kinds of density that make sense in sufficiently well-behaved semigroups including $(\mathbb {N},+)$ and $(\mathbb {N},\cdot )$. It was recently shown that sets in $\mathbb {N}$ which are multiplicatively large must contain arbitrarily large geoarithmetic progressions, that is, sets of the form $\big \{r^j(a\!+\!id)\!:i,j\in \{0,1,\dotsc ,k\}\big \}$, as well as sets of the form $\big \{b(a+id)^j:i,j\in \{0,1,\dotsc ,k\}\big \}$. Consequently, given a finite partition of $\mathbb {N}$, one cell must contain such configurations. In the partition case we show that we can get substantially stronger conclusions. We establish some combined additive and multiplicative Ramsey theoretic consequences of known algebraic results in the semigroups $(\beta \mathbb {N},+)$ and $(\beta \mathbb {N},\cdot )$, derive some new algebraic results, and derive consequences of them involving geoarithmetic progressions. For example, we show that given any finite partition of $\mathbb {N}$ there must be, for each $k$, sets of the form $\big \{b(a+id)^j:i,j\in \{0,1,\dotsc ,k\}\big \}$ together with $d$, the arithmetic progression $\big \{a+id:i\in \{0,1,\dotsc ,k\}\big \}$, and the geometric progression $\big \{bd^j:j\in \{0,1,\dotsc ,k\}\big \}$ in one cell of the partition. More generally, we show that, if $S$ is a commutative semigroup and ${\mathcal F}$ a partition regular family of finite subsets of $S$, then for any finite partition of $S$ and any $k\in \mathbb {N}$, there exist $b,r\in S$ and $F\in {\mathcal F}$ such that $rF\cup \{b(rx)^j:x \in F,j\in \{0,1,2,\ldots ,k\}\}$ is contained in a cell of the partition. Also, we show that for certain partition regular families ${\mathcal F}$ and ${\mathcal G}$ of subsets of $\mathbb {N}$, given any finite partition of $\mathbb {N}$ some cell contains structures of the form $B \cup C \cup B\cdot C$ for some $B\in {\mathcal F}, C\in {\mathcal G}$.