In this paper we study some additive properties of subsets of the set $\nats$ of positive integers: A subset $A$ of $\nats$ is called { \it $k$-summable} (where $k\in\ben$) if $A$ contains $\textstyle \big{ \sum_{ n\in F}x_n | \emp\neq F\subseteq { 1,2,...,k\} \big}$ for some $k$-term sequence of natural numbers $x_1<x_2 < ... < x_k$. We say $A \subseteq \nats$ is finite FS-big if $A$ is $k$-summable for each positive integer $k$. We say is $A \subseteq \nats$ is infinite FS-big if for each positive integer $k,$ $A$ contains ${ \sum_{ n\in F}x_n | \emp\neq F\subseteq \nats and #F\leq k}$ for some infinite sequence of natural numbers $x_1<x_2 < ... $. We say $A\subseteq \nats $ is an IP-set if $A$ contains ${ \sum_{ n\in F}x_n | \emp\neq F\subseteq \nats and #F<\infty}$ for some infinite sequence of natural numbers $x_1<x_2 < ... $. By the Finite Sums Theorem [5], the collection of all IP-sets is partition regular, i.e., if $A$ is an IP-set then for any finite partition of $A$, one cell of the partition is an IP-set. Here we prove that the collection of all finite FS-big sets is also partition regular. Let $\TM =011010011001011010... $ denote the Thue-Morse word fixed by the morphism $0\mapsto 01$ and $1\mapsto 10$. For each factor $u$ of $\TM$ we consider the set $\TM\big|_u\subseteq \nats$ of all occurrences of $u$ in $\TM$. In this note we characterize the sets $\TM\big|_u$ in terms of the additive properties defined above. Using the Thue-Morse word we show that the collection of all infinite FS-big sets is not partition regular.
Categories: math.CO math.GN