Big Ramsey Degrees of Countable Ordinals

2023/05/12 by Boyland, Joanna, Gasarch, William, Hurtig, Nathan +1
#05D10 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2305.07192

Abstract

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 ∈ 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'⊆ 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 ∈ N, T(a,α) is finite. We find exact value of T(a,α) for all ordinals less than ω^ω and all a∈ N.

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