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- W4289763708 abstract "The Hales-Jewett Theorem states that given any finite nonempty set $A$ and any finite coloring of the free semigroup $S$ over the alphabet $A$ there is a {it variable word/} over $A$ all of whose instances are the same color. This theorem has some extensions involving several distinct variables occurring in the variable word. We show that, when combined with a sufficiently well behaved homomorphism, the relevant variable word simultaneously satisfies a Ramsey-Theoretic conclusion in the other structure. As an example we show that if $tau$ is the homomorphism from the set of variable words into the natural numbers which associates to each variable word $w$ the number of occurrences of the variable in $w$, then given any finite coloring of $S$ and any infinite sequence of natural numbers, there is a variable word $w$ whose instances are monochromatic and $tau(w)$ is a sum of distinct members of the given sequence. Our methods rely on the algebraic structure of the Stone-v Cech compactification of $S$ and the other semigroups that we consider. We show for example that if $tau$ is as in the paragraph above, there is a compact subsemigroup $P$ of $betaben$ which contains all of the idempotents of $betaben$ such that, given any $pin P$, any $Ain p$, and any finite coloring of $S$, there is a variable word $w$ whose instances are monochromatic and $tau(w)in A$. We end with a new short algebraic proof of an infinitary extension of the Graham-Rothschild Parameter Sets Theorem." @default.
- W4289763708 created "2022-08-04" @default.
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- W4289763708 date "2018-07-04" @default.
- W4289763708 modified "2023-09-26" @default.
- W4289763708 title "Combining extensions of the Hales-Jewett Theorem with Ramsey Theory in other structures" @default.
- W4289763708 doi "https://doi.org/10.48550/arxiv.1807.01485" @default.
- W4289763708 hasPublicationYear "2018" @default.
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