Hanf numbers for poset games

Abstract

Given two well partial orders $(P;\leq_P)$ and $(T;\leq_T)$, each with a minimum element, we study the following question: which player has a winning strategy for Chomp on the poset $(P\times T;\leq_{P\times T})$? Here, $(P\times T;\leq_{P\times T})$ denotes the poset obtained as the Cartesian product of $P$ and $T$, equipped with the corresponding lexicographic order. The answer to this very natural question depends strongly on the specific choice of $(P;\leq_P)$ and $(T;\leq_T)$. For this reason, we restrict our attention to classes of posets given by powers of a fixed poset: $\{(P^σ;\leq_{P^σ})\mid σ\in\mathrm{Ord}\}$. A fundamental fact about these classes of structures is that, if the second player does not have a winning strategy for all the posets in $\{(P^σ;\leq_{P^σ})\mid σ\in\mathrm{Ord}\}$, there exists an ordinal $ξ$ such that the second player has a winning strategy on $(P^ξ;\leq_{P^ξ})$ but not on $(P^γ;\leq_{P^γ})$ for all $γ\geqξ+1$. Determining the corresponding ordinal for this Hanf number-style property constitutes the main objective of this work. Inspired by results of García-Marco and Knauer, we focus on classes of posets with a purely algebraic definition. These posets arise from submonoids (with respect to the natural sum, or Hessenberg sum) of ordinals of the form $ω^σ$ and are generated by sets of ordinals. In the process, we provide a test to determine whether a finite set $Γ$ of ordinals indeed yields well partial orders, and, using set-theoretic techniques, we establish an upper bound for the ordinal $ξ$: if $Γ\subsetω$, then $ξ<ω_1$, and otherwise $ξ<|\bigcupΓ|^+$.