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Statement

Claim (§3, p. 12, unnumbered). For the template $\underbrace{1\ldots1}_{r}\underbrace{2\ldots2}_{s}$ over $[2]$ and every $k$, there is an $n$ such that every $k$-coloring of $[2]^n$ contains a monochromatic block set with that template whose blocks are $r+s$ singletons. The block set is uniform, of degree $r+s$, so Conjectures E and F (see the conjecture page) hold for $m=2$; the paper calls the case $m=1$ trivial (p. 12).

Proof sketch

P. 12. Choose $n$ by Ramsey's theorem so that every $k$-coloring of the $s$-subsets of $[n]$ has a homogeneous set of size $r+s$. Color an $s$-set by the word that is $2$ on it and $1$ elsewhere, take the homogeneous set's points as singleton blocks, and fix $1$ outside it.

Note

This case is what covers alphabets of at most two letters in Proposition 2.4, where Lemma 2.3 needs $m\ge3$. When $r=0$ or $s=0$ the template has one rearrangement and any $r+s$ singleton blocks serve.

Source. Imre Leader, Paul A. Russell and Mark Walters, Transitive sets in Euclidean Ramsey theory, J. Combin. Theory Ser. A 119 (2012), no. 2, 382--396, doi:10.1016/j.jcta.2011.09.005; page from the arXiv version 1012.1350v1 identified in the source digest.

Read depth. Claims checked: the argument (p. 12) was read in full and followed.

Bears on

• Problem 174: settles the two-letter cases of the paper's block-set conjectures, which on their own prove no new set Ramsey.