Transforming rectangles into squares, with applications to strong colorings

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

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

• every positive integer $n$,
• every coloring $d:n\times n\to {\lambda }^{+}$, 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.