Big Ramsey Degrees of Countable Ordinals
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912696451989504 |
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| author | Boyland, Joanna Gasarch, William Hurtig, Nathan Rust, Robert |
| author_facet | Boyland, Joanna Gasarch, William Hurtig, Nathan Rust, Robert |
| contents | Ramsey's theorem states that for all finite colorings of an infinite set, there exists an infinite homogeneous subset. What if we seek a homogeneous subset that is also order-equivalent to the original set? Let $S$ be a linearly ordered set and $a \in N$. The big Ramsey degree of $a$ in $S$, denoted $T(a,S)$, is the least integer $t$ such that, for any finite coloring of the $a$-subsets of $S$, there exists $S'\subseteq S$ such that (i) $S'$ is order-equivalent to $S$, and (ii) if the coloring is restricted to the $a$-subsets of $S'$ then at most $t$ colors are used.
Mašulović \& Šobot (2019) showed that $T(a,ω+ω)=2^a$. From this one can obtain $T(a,ζ)=2^a$. We give a direct proof that $T(a,ζ)=2^a$.
Mašulović and Šobot (2019) also showed that for all countable ordinals $α< ω^ω$, and for all $a \in N$, $T(a,α)$ is finite. We find exact value of $T(a,α)$ for all ordinals less than $ω^ω$ and all $a\in N$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_07192 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Big Ramsey Degrees of Countable Ordinals Boyland, Joanna Gasarch, William Hurtig, Nathan Rust, Robert Combinatorics 05D10 Ramsey's theorem states that for all finite colorings of an infinite set, there exists an infinite homogeneous subset. What if we seek a homogeneous subset that is also order-equivalent to the original set? Let $S$ be a linearly ordered set and $a \in N$. The big Ramsey degree of $a$ in $S$, denoted $T(a,S)$, is the least integer $t$ such that, for any finite coloring of the $a$-subsets of $S$, there exists $S'\subseteq S$ such that (i) $S'$ is order-equivalent to $S$, and (ii) if the coloring is restricted to the $a$-subsets of $S'$ then at most $t$ colors are used. Mašulović \& Šobot (2019) showed that $T(a,ω+ω)=2^a$. From this one can obtain $T(a,ζ)=2^a$. We give a direct proof that $T(a,ζ)=2^a$. Mašulović and Šobot (2019) also showed that for all countable ordinals $α< ω^ω$, and for all $a \in N$, $T(a,α)$ is finite. We find exact value of $T(a,α)$ for all ordinals less than $ω^ω$ and all $a\in N$. |
| title | Big Ramsey Degrees of Countable Ordinals |
| topic | Combinatorics 05D10 |
| url | https://arxiv.org/abs/2305.07192 |