An infinitary extension of the Graham–Rothschild Parameter Sets Theorem

Abstract:

The Graham-Rothschild Parameter Sets Theorem is one of the most powerful results of Ramsey Theory. (The Hales-Jewett Theorem is its most trivial instance.) Using the algebra of $\beta S$, the Stone-Čech compactification of a discrete semigroup, we derive an infinitary extension of the Graham-Rothschild Parameter Sets Theorem. Even the simplest finite instance of this extension is a significant extension of the original. The original theorem says that whenever $k<m$ in $\mathbb {N}$ and the $k$-parameter words are colored with finitely many colors, there exist a color and an $m$-parameter word $w$ with the property that whenever a $k$-parameter word of length $m$ is substituted in $w$, the result is in the specified color. The “simplest finite instance” referred to above is that, given finite colorings of the $k$-parameter words for each $k<m$, there is one $m$-parameter word which works for each $k$. Some additional Ramsey Theoretic consequences are derived. We also observe that, unlike any other Ramsey Theoretic result of which we are aware, central sets are not necessarily good enough for even the $k=1$ and $m=2$ version of the Graham-Rothschild Parameter Sets Theorem.