Ageron 2021 new lower bounds schur weak schur

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corollary_2_9: The exponential lower bound on Schur numbers and on the multicolor Ramsey numbers of the triangle that follows from the recursion S(n+5) at least 380 S(n) + 148; the best lower growth rate the site records for f(k).

inequality_6: The recursive lower bound for Schur numbers produced by the paper's best S-template with six colors, one of the paper's three new inequalities, its template found with a SAT solver; the source of the growth rate 380 to the one fifth.

theorem_2_3: The S-template composition theorem, the paper's rephrasing of Rowley's construction in terms of Schur numbers, with its Corollary 2.4, S(n+k) at least S+(n+1) S(k) + m_{n+1} − 1, from which the paper's recursions (2)--(7) for Schur numbers come.

theorem_3_1: A weakly sum-free n-partition of [1, q] and a sum-free k-partition of [1, p] give a weakly sum-free (n+k)-partition of [1, p(q + ⌈q/2⌉ + 1) + q]; its Corollary 3.2 bounds WS(n+k) below by S(k) and WS(n).

theorem_3_17: The paper's main result, the weak Schur template theorem: a b-WS-template of width a with n+1 colors and a sum-free k-partition of [1, p] give a partition of [1, pa + b] into n+k weakly sum-free sets; with Corollary 3.18 and the paper's weak Schur recursions (10) and (11).

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Romain Ageron, Paul Casteras, Thibaut Pellerin, Yann Portella, Arpad Rimmel, Joanna Tomasik, New lower bounds for Schur and weak Schur numbers. arXiv:2112.03175 (2021).

The authors formalize Rowley's template-based constructions for sum-free partitions as S-templates, introduce the auxiliary sequence S+(n) (Proposition 2.2), and prove product-type recursions: Theorem 2.3 and Corollary 2.4 combine an S-template of width q on n+1 colors with a sum-free partition of length p to build partitions witnessing larger Schur numbers, with a variant in Theorem 2.6 and Corollary 2.7. New templates found by search give the inequality S(n+5) >= 380S(n) + 148, hence the growth-rate bound gamma >= 380^(1/5) ~ 3.28 for Schur numbers and for the multicolor Ramsey numbers R_n(3) (Corollary 2.9). Section 3 generalizes templates to the weakly sum-free setting, giving inequalities such as Corollary 3.2 (WS(n+k) >= S(k)(WS(n) + ceil(WS(n)/2) + 1) + WS(n)), the more general Theorem 3.17 and its corollaries, and the inequalities (10) WS(n+3) >= 42S(n) + 24, found with a SAT solver, and (11) WS(n+4) >= 132S(n) + 26, obtained by combining an S-template of width 33 with a WS-template of width 4 (the best WS-template found by computer search gives WS(n+4) >= 127S(n) + 68). The explicit partitions yield the new records S(9) >= 17803, S(10) >= 60948, S(11) >= 203828, S(12) >= 644628 and WS(6) >= 646, WS(9) >= 22536, WS(10) >= 71256, WS(11) >= 243794, WS(12) >= 815314. The Schur-side results (Table 1, inequality (6), Corollary 2.9) are what bears on problem 183 (multicolor Ramsey numbers of the triangle, through S(n) <= R_n(3) - 2) and on problem 483, whose f(k) is the strong Schur number S(k) + 1 (the site's equation a + b = c allows a = b, and the paper's Definition 1.1 is the same convention); the weak Schur numbers WS(n), which require a != b, are not the quantity of problem 483.

Source: https://arxiv.org/abs/2112.03175.

The copy read for this card is the arXiv v2 of 4 April 2022 (dated "April 4, 2022" on p. 1; twenty pages, printed page equals PDF page); the arXiv listing read shows v1 of 6 December 2021 and v2, and no journal reference; a Crossref bibliographic query the same day found no journal record, so the paper is cited as a preprint. Read status: claims checked for Definitions 1.1--1.4, 2.1, 3.4, 3.9 and 3.13--3.15, Propositions 2.2 and 3.16, Theorems 2.3, 2.6, 3.1, 3.17 and 3.21, Corollaries 2.4, 2.7, 2.9, 3.2, 3.18 and 3.22, displays (2)--(11) and Tables 1--4, read clause by clause on the page images of pp. 1--16; the proofs of Theorems 2.3, 3.1 and 3.17 were followed. The templates of Appendices A and B that realize (4)--(6) and (10), and the partition of Appendix C behind $WS(6)\ge646$, were not inspected. Nothing here is independently reviewed. Result pages: theorem_2_3, inequality_6, corollary_2_9, theorem_3_1, theorem_3_17. The preprint carries no arXiv stamp and prints no notice; the arXiv abstract page (https://arxiv.org/abs/2112.03175v2, read 2026-10-02) names arXiv's non-exclusive distribution license, every other right reserved.

Bears on. #183: inequality (6) (p. 6), $S(n+5)\ge380S(n)+148$, and Corollary 2.9 (p. 7), "The growth rate for Schur numbers (and Ramsey numbers $R_n(3)$) satisfies $\gamma\ge\sqrt[5]{380}\approx3.28$", whose proof passes to $R_n(3)$ through $S(n)\le R_n(3)-2$: a lower bound on the problem's limit, $\liminf R(3;k)^{1/k}\ge380^{1/5}$; the paper does not treat the upper side. With $S(n)\le R_n(3)-2$, the abstract's $S(9)\ge17\,803$ and $S(10)\ge60\,948$ give $R_9(3)\ge17\,805$ and $R_{10}(3)\ge60\,950$, an arithmetic step of this card, not stated in the paper. #483: the problem's $f(k)$ is $S(k)+1$; iterating inequality (6) from the five known values gives $f(k)\ge c\cdot380^{k/5}$ with an absolute $c>0$ (about $0.379$, computed on the corollary's page), which is the source of the site's lower bound, and Corollary 2.9 gives the growth rate $380^{1/5}$. The site's display $(380)^{k/5}-O(1)\le f(k)$ is stronger than what the paper proves. Tables 1 and 3 give $S(9)\ge17\,803$ to $S(12)\ge644\,628$, that is $f(9)\ge17\,804$ to $f(12)\ge644\,629$. The weak Schur results of Section 3 bear on neither problem.

Contents.

• Theorem 2.3 (p. 3) and Corollary 2.4 (p. 5): an S-template of width $q$ with $n+1$ colors and a sum-free $k$-partition of $[\![1,p]\!]$ give a sum-free $(n+k)$-partition of $[\![1,pq+m_{n+1}-1]\!]$, hence $S(n+k)\ge S^+(n+1)S(k)+m_{n+1}-1$; the paper's rephrasing of Rowley's construction.
• Displays (2)--(7) (p. 6): the recursions $S(n+j)\ge aS(n)+b$ for $j=1,\ldots,6$; (5) $S(n+4)\ge111S(n)+43$, (6) $S(n+5)\ge380S(n)+148$ and (7) $S(n+6)\ge1160S(n)+536$ are the paper's own.
• Corollary 2.9 (p. 7): the growth rate of $S(n)$ and of $R_n(3)$ is at least $380^{1/5}\approx3.28$.
• Theorem 3.1 (p. 8) and Corollary 3.2 (p. 9): a weakly sum-free $n$-partition of $[\![1,q]\!]$ and a sum-free $k$-partition of $[\![1,p]\!]$ give a weakly sum-free $(n+k)$-partition of $[\![1,p(q+\lceil q/2\rceil+1)+q]\!]$, hence $WS(n+k)\ge S(k)(WS(n)+\lceil WS(n)/2\rceil+1)+WS(n)$.
• Theorem 3.17 (p. 12) and Corollary 3.18 (p. 14): the weak Schur template theorem, a $b$-WS-template of width $a$ with $n+1$ colors and a sum-free $k$-partition of $[\![1,p]\!]$ give a weakly sum-free $(n+k)$-partition of $[\![1,pa+b]\!]$, hence $WS(n+k)\ge S(k)WS^+(n+1)+b_{\max}$; Theorem 3.1 is a special case.
• Displays (8)--(11) (p. 15): $WS(n+1)\ge4S(n)+2$ and $WS(n+2)\ge13S(n)+8$ (Rowley), and the paper's $WS(n+3)\ge42S(n)+24$ and $WS(n+4)\ge132S(n)+26$.
• Tables 1 and 2 (p. 2), with Tables 3 (p. 7) and 4 (p. 16): the new lower bounds $S(9)\ge17\,803$, $S(10)\ge60\,948$, $S(11)\ge203\,828$, $S(12)\ge644\,628$, and $WS(6)\ge646$, $WS(9)\ge22\,536$, $WS(10)\ge71\,256$, $WS(11)\ge243\,794$, $WS(12)\ge815\,314$.

Results.

• Theorem 2.3 (p. 3), with Corollary 2.4 (p. 5): the S-template composition theorem.
• Inequality (6) (p. 6): $S(n+5)\ge380S(n)+148$.
• Corollary 2.9 (p. 7): the growth rate $380^{1/5}$ for $S(n)$ and $R_n(3)$.
• Theorem 3.1 (p. 8), with Corollary 3.2 (p. 9): the weak Schur analogue of Abbott and Hanson's construction.
• Theorem 3.17 (p. 12), with Corollary 3.18 (p. 14) and displays (10) and (11) (p. 15): the weak Schur template theorem.

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