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Statement

Definitions (p. 2). A certificate $S(k,n)$ is a $k$-coloring of $1,\ldots,n$ with no monochromatic solution of $a+b=c$ for $1\le a,b,c\le n$; an extreme certificate is one of maximum size, so its size is $S(k)$. The modular Schur number $S_{\mathrm{mod}}(k)$ is the largest $n$ for which some $k$-coloring of $1,\ldots,n$ has no monochromatic solution of $a+b\equiv c\pmod{n+1}$ with $1\le a,b,c\le n$. The palindromic Schur number $S_{\mathrm{pd}}(k)$ adds the condition that "the numbers $i$ and $n+1-i$ with $1\leq i\leq n/2$ have the same color—except in case $2i=n+1-i$" (p. 2). The paper records $S(k)\ge S_{\mathrm{mod}}(k)$ and $S_{\mathrm{mod}}(k)\ge S_{\mathrm{pd}}(k)$ for every $k$ (p. 2).

Values for five colors (p. 2, with p. 6). From $S(5)=160$ (main result) and the palindromic certificate $S(5,160)$ of Figure 1 (p. 3), which the paper states is also an extreme certificate, $S(5)=S_{\mathrm{mod}}(5)=S_{\mathrm{pd}}(5)=160$. The paper says its result implies $S(5)=S_{\mathrm{mod}}(5)$, conjectured by Abbott and Wang (1977) for all $k$, and calls the equality with $S_{\mathrm{pd}}(5)$ "a new result for $k=5$" (p. 2); p. 6 restates it as $S(5)=S_{\mathrm{pd}}(5)=160$.

Enumeration (p. 7, and the contributions list on p. 1). There are exactly $2\,447\,113\,088$ extreme certificates $S(5,160)$. Of these, $315\,853\,824$ are modular, and of the modular ones $334\,752$ are palindromes. The paper notes that the last number had been conjectured before by Fredricksen and Sweet (2000), and in footnote 3 (p. 7) that Fredricksen and Sweet state the number of palindromic extreme certificates $S(5,160)$ as $309\,408$, whereas its method produces $334\,752$. The contributions list (p. 1) describes the count as all five-colorings of $1$ to $160$ without a monochromatic $a+b=c$; the computation on p. 7 runs on the formula to which the color-symmetry-breaking predicates of pp. 3--4 have been added, and the paper does not say in so many words whether the count is taken modulo permutation of the colors.

Source. M. J. H. Heule, Schur Number Five, arXiv:1711.08076v1 (21 November 2017), nine pages without printed page numbers; locators are PDF pages. The edition read is identified on the source card.

Read depth. Claims checked: the paragraph "Schur Numbers and Variants" (p. 2), the caption of Figure 1 and the definitions of certificates (pp. 2--3), the section "No Backbone, but Backdoors" (pp. 6--7) and footnote 3 were read clause by clause on the page images. The counts rest on a computation the paper describes; nothing was recomputed here, and unlike the upper bound $S(5)\le160$ the enumeration is not stated to be covered by the certified proof.

Proof pointer

Section "No Backbone, but Backdoors" (pp. 6--7): for each top-level cube under which the five-color formula for $n=160$ with symmetry breaking is satisfiable, the backbone is computed and, where needed, extended by look-aheads into a backdoor; the resulting $1616$ backdoors cover all satisfying assignments, and a model counter (sharpSAT) counts the extreme certificates from them. The values of $S_{\mathrm{mod}}(5)$ and $S_{\mathrm{pd}}(5)$ follow from the chain of inequalities above, the palindromic certificate of Figure 1 and $S(5)\le160$.

Dependencies

Main result $S(5)=160$; the palindromic certificate of Figure 1 (p. 3).

Bears on

No problem page of the corpus consumes this result. The modular and palindromic Schur numbers are variants the paper treats; the Schur number itself, the subject of Problem 483, is covered by the main result.