Transforming rectangles into squares, with applications to strong colorings

Abstract: It is proved that every singular cardinal  $\lambda$ admits a function $\textbf{rts}:[\lambda^+]^2\rightarrow[\lambda^+]^2$ that transforms rectangles into squares.
That is, whenever $A,B$ are cofinal subsets of $\lambda^+$, we have $\textbf{rts}[A\circledast B]\supseteq C\circledast C$, for some cofinal subset $C\subseteq\lambda^+$.

As a corollary, we get that for every uncountable cardinal $\lambda$, the classical negative partition relation $\lambda^+\nrightarrow[\lambda^+]^2_{\lambda^+}$ coincides with the following higher arity statement. There exists a function $c:[\lambda^+]^2\rightarrow\lambda^+$ such that for

• every positive integer $n$,
• every coloring $d:n\times n\rightarrow\lambda^+$, and
• every family $\mathcal A\subseteq[\lambda^+]^n$ of size $\lambda^+$ of mutually disjoint sets,

there exist $a,b\in\mathcal A$ with $\max(a)<\min(b)$ such that $$c(a_i,b_j)=d(i,j)\text{ for all }i,j<n.$$(here, $a_i$ denotes the $i_{th}$-element of $a$, and $b_j$ denotes the $j_{th}$-element of $b$.)

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Citation information:

A. Rinot, Transforming rectangles into squares, with applications to strong colorings, Adv. Math., 231(2): 1085-1099, 2012.