Soukup 2015 sums anti ramsey colourings reals

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corollary_2_2: Under the continuum hypothesis there is a coloring of the reals with continuum many colors that takes every color on the sums of N distinct elements of every uncountable set, for every N at least two.

corollary_2_3: It is consistent that the number of colors in the Hindman-Leader-Strauss coloring of the reals cannot be increased to three, by Shelah's consistency of a positive square-bracket relation for pairs with three colors.

corollary_3_2: A two-coloring of the reals, in ZFC, under which every uncountable set has sums of N distinct elements in both colors for every N at least two; with N equal to two this is the negative answer to Problem 965 without the continuum hypothesis.

lemma_2_1: For a cardinal nu, a nu-coloring of the finite subsets of the Cantor set realizing every color on unions of N distinct members of every uncountable family exists exactly when a nu-coloring of the reals realizing every color on N-fold sums of every uncountable set exists, and either gives a negative square-bracket relation for pairs.

theorem_3_1: A two-coloring of the finite subsets of the Cantor set, in ZFC, under which every uncountable family and every N at least two admit N distinct members whose union has either color; the union form of the anti-Ramsey coloring of the reals.

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Dániel T. Soukup and William Weiss, Sums and anti-Ramsey colourings of $\mathbb R$. A five-page manuscript with no date in its text, no arXiv identifier and no journal; the authors' addresses are the Alfréd Rényi Institute of Mathematics and the University of Toronto. Its reference list cites Hindman, Leader and Strauss "(to appear in Abh. Math. Sem. Univ. Hamburg)" and Komjáth, A certain 2-coloring of the reals, "to appear in ???", so the text predates both publications (Komjáth's in 2016, the Hindman--Leader--Strauss paper in 2017). The site's Problem 965 page cites it in its commentary as "Soukup and Weiss" with no key; the formal-conjectures file for that problem cites it as [SWCol] from the first author's site (danieltsoukup.github.io/academic/finset_colouring.pdf).

Edition read. The copy read for this card is the manuscript as typeset by the authors (pdfTeX; its metadata gives a creation date of 7 September 2015), five pages with a complete text layer, read in the text layer and on the rendered pages 1--5. Provenance: the repository's survey download set (the download URL recorded when the copy was obtained is http://www.renyi.hu/~dsoukup/sums.pdf); 282,304 bytes. The manuscript prints no copyright or license line on its first two or last two pages; its recorded source (http://www.renyi.hu/~dsoukup/sums.pdf) is on the author's site, whose page returned HTTP 404 on 2026-10-02 and so states no terms, and no publisher page exists for the unpublished text; the term is unstated.

Publication status. A Crossref bibliographic query for the title and authors, arXiv author and abstract searches, and the citation lists of the Hindman--Leader--Strauss paper and of Komjáth's paper in Semantic Scholar returned no record of it. The manuscript's own sentence on its result (p. 1): "The same result was independently proved by P. Komjáth [2]." The card records that sentence as the source's own and claims no independent check of the argument.

Read status: claims checked for the abstract, Theorem 1.1, Lemma 2.1, Corollaries 2.2 and 2.3, Theorem 3.1, Corollary 3.2 and Theorem 4.1, read clause by clause in the text layer and on the rendered pages 1--5; the proof of Theorem 3.1 (pp. 3--4), the proof of Lemma 2.1 (p. 2) and the short proofs of Corollaries 2.2 (p. 2) and 2.3 (p. 3) were read for their structure and not checked step by step; nothing here is independently reviewed.

Contents

• Abstract (p. 1): the abstract recalls that Hindman, Leader and Strauss [1] proved under CH that some coloring $F:\mathbb R\to2$ has $F''\{x+y:x\ne y\in X\}=2$ for every uncountable $X\subseteq\mathbb R$, and states the manuscript's result: the same holds in ZFC, answering a problem posed in [1].
• Section 1, Introduction (p. 1): Theorem 1.1, the Hindman--Leader--Strauss result as the manuscript restates it: assuming the Continuum Hypothesis there is a coloring $F:\mathbb R\to2$ that "is not monochromatic on any set $\{\sum E:E\in[X]^N\}$" whenever $X\subseteq\mathbb R$ is uncountable and $N\ge1$ (printed with a stray "$=2$" after the set). The introduction then poses the question, which it says also appears in [1], whether CH can be dropped from Theorem 1.1, announces that it can, records that "The same result was independently proved by P. Komjáth [2]", and says the manuscript also discusses whether three or more colors are possible.
• Section 2, Sums versus unions (pp. 1--3): Lemma 2.1, for a cardinal $\nu$: (1) some $f:[2^\omega]^{<\omega}\to\nu$ has, for every uncountable $X\subseteq[2^\omega]^{<\omega}$, every $N\ge2$ and each color $i<\nu$, $N$ distinct members $a_0,\dots,a_{N-1}$ of $X$ with $f(\bigcup_{j<N}a_j)=i$; (2) there is a coloring $F:\mathbb R\to\nu$ with $F''\{\sum E:E\in[X]^N\}=\nu$ for any uncountable $X\subseteq\mathbb R$ and $N\in\omega\setminus2$; (3) $2^{\aleph_0}\not\to[\omega_1]^2_\nu$; then (1) $\Leftrightarrow$ (2) $\Rightarrow$ (3), through a basis of $\mathbb R$ over $\mathbb Q$ and supports ("(1) $\Rightarrow$ (2) was essentially proved in [1]"). Corollary 2.2: CH implies such an $F$ with $2^\omega$ colors, by Lemma 5.2.6 of Todorcevic's book, "clearly the best possible". Corollary 2.3: consistently the number of colors in Theorem 1.1 cannot be increased to three, by Shelah's consistency of $2^{\aleph_0}\to[\omega_1]^2_3$.
• Section 3, A 2-colouring in ZFC (pp. 3--4): Theorem 3.1, statement (1) of Lemma 2.1 with $\nu=2$, proved with the Sierpiński coloring applied to the pair of a finite set that realizes its maximal splitting level in $2^{\le\omega}$, after normalizing an uncountable family to a $\Delta$-system with coordinatewise monotone branches, the case $N=2$ first and then general $N$; Corollary 3.2: a coloring $F:\mathbb R\to2$ with $F''\{\sum E:E\in[X]^N\}=2$ for any uncountable $X\subseteq\mathbb R$ and $N\in\omega\setminus2$.
• Section 4, Open problems (pp. 4--5): Theorem 4.1, quoted from [1]: if $2^\omega<\aleph_\omega$ there is a finite coloring of $\mathbb R$ not constant on any $X+X$ with $X\subseteq\mathbb R$ infinite; p. 5 adds "It is not known if the cardinal arithmetic assumption can be removed from this result."
• References (p. 5): [1] Hindman, Leader and Strauss (to appear); [2] Komjáth (to appear); [3] Shelah, Was Sierpinski right? I, Israel J. Math. 62 (1988), 355--380; [4] Todorcevic, Walks on Ordinals and Their Characteristics (2007).

Compiled scope

Lemma 2.1, Corollaries 2.2 and 2.3, Theorem 3.1 and Corollary 3.2 are compiled as statements with proof pointers; Theorems 1.1 and 4.1, which the manuscript quotes from [1], are recorded above only. No proof was reconstructed or checked.

Bears on. #965: Corollary 3.2 (p. 4) with $N=2$ states, in ZFC, a two-coloring $F$ of $\mathbb R$ under which every uncountable $X\subseteq\mathbb R$, in particular every $X$ of size $\aleph_1$, has sums $x+y$ of distinct elements in both colors, which contradicts the problem's statement; it comes from Theorem 3.1 (p. 3) through Lemma 2.1 (pp. 1--2) with $\nu=2$. Corollary 2.2 (p. 2) gives the same with $2^\omega$ colors under CH. Corollary 2.3 (pp. 2--3) concerns three colors only: it is consistent that no three-coloring of $\mathbb R$ takes every color on the $N$-fold sums of every uncountable set for every $N\ge2$; it gives no monochromatic set of sums. The manuscript is unpublished; it states that Komjáth proved the same result independently.

Results. Labels and pages are the print's.

• Lemma 2.1 (p. 1; proof p. 2): for a cardinal $\nu$, the union form (1) and the sum form (2) of a $\nu$-coloring taking every color on every uncountable set are equivalent, and imply $2^{\aleph_0}\not\to[\omega_1]^2_\nu$.
• Corollary 2.2 (p. 2): under CH some $F:\mathbb R\to2^\omega$ has $F''\{\sum E:E\in[X]^N\}=2^\omega$ for any uncountable $X\subseteq\mathbb R$ and $N\in\omega\setminus2$.
• Corollary 2.3 (p. 2; proof p. 3): consistently the number of colors in Theorem 1.1 cannot be increased to three.
• Theorem 3.1 (p. 3): some map $f:[2^\omega]^{<\omega}\to2$ has, for every uncountable $X\subseteq[2^\omega]^{<\omega}$, every $N\ge2$ and each color $i<2$, $N$ distinct members $a_0,\dots,a_{N-1}$ of $X$ with $f(\bigcup_{j<N}a_j)=i$.
• Corollary 3.2 (p. 4): there is a coloring $F:\mathbb R\to2$ such that $F''\{\sum E:E\in[X]^N\}=2$ for any uncountable $X\subseteq\mathbb R$ and $N\in\omega\setminus2$.

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