Problem 483

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Statement. Let $f(k)$ be the minimal $N$ such that if $\{1,\ldots,N\}$ is $k$-coloured then there is a monochromatic solution to $a+b=c$. Estimate $f(k)$. In particular, is it true that $f(k) < c^k$ for some constant $c>0$?

Formulation. The site's wording, accessed 2026-09-18 (page last edited 1 September 2026). The literature's Schur number $S(k)$ is the largest $N$ that admits a $k$-coloring of $\{1,\ldots,N\}$ with no monochromatic solution of $a+b=c$, where $a=b$ is allowed; the site's $f(k)$ is the least $N$ that forces one, so $f(k)=S(k)+1$ (Heule, footnote 1; Fredricksen and Sweet, footnote 1; both quoted on their cards). This page uses that shift throughout: the exact values $S(1)=1$, $S(2)=4$, $S(3)=13$, $S(4)=44$, $S(5)=160$ are the site's $f(1)=2$, $f(2)=5$, $f(3)=14$, $f(4)=45$, $f(5)=161$. The weak Schur numbers $WS(k)$, which require $a\ne b$, are a different quantity and are not this problem. The question "$f(k)<c^k$" asks whether $f$ grows at most exponentially; since $f$ has growth rate at least $380^{1/5}$ (Ageron et al.'s Corollary 2.9, below), any such $c$ is at least $380^{1/5}\approx3.28$. The request to estimate $f(k)$ is read as Erdős's 1961 item poses it: as the growth question whether $f(k)<c^k$ for some $c$, and whether $f(k)^{1/k}$ tends to a finite limit. The results the site credits settle neither question, so none has a claim page. They are the lower bound of Ageron et al. (growth rate at least $380^{1/5}$) with the earlier bases of Exoo and of Fredricksen and Sweet; the upper bound $(e-\tfrac16)k!$ of Xu, Xie and Chen, with the earlier bounds of Whitehead and of Wan and Eliahou's English proof; and Heule's exact value $f(5)=161$.

Erdős's own wording, 1961 (printed pp. 232--233): "SCHUR proved that if we split the integers $<en!$ into $n$ classes the equation $x+y=z$ is always solvable in integers of the same class. Denote by $f(n)$ the smallest integer with this property. It seems likely that $f(n)$ is very much less than $en!$, in fact it has been conjectured that $f(n)<c^n$ and $f(n)^{1/n}\to C$." The site's $f(k)$ is this $f(n)$ less one (as the two are worded, Erdős's $f(n)$ is the least $N$ for which every split of the integers below $N$ into $n$ classes has a solution in one class, the site's $f(k)$ the least $N$ for $\{1,\ldots,N\}$), and its question is this conjecture; the item begins on p. 232 although the site cites p. 233. The 1965 survey (printed p. 188) states the inverse form: with $H(n)$ the least number of sum-free classes into which $\{1,\ldots,n\}$ can be split, "Schur [8] proved that $H(cn!)>n$. It seems very hard to decide whether $H(n)>c\log n$ holds for a certain $c>0$."

Status. OPEN, the site's label. The bounds in hand are a growth rate of at least $380^{1/5}$ below and $f(k)\le(e-\tfrac16)\,k!$ for $k\ge4$ above: the lower bound is Corollary 2.9 of Ageron, Casteras, Pellerin, Portella, Rimmel and Tomasik (2021--22, an arXiv preprint), from their recursion $S(n+5)\ge380S(n)+148$; the upper bound is Xu, Xie and Chen's $R_k(3)\le k!(e-1/6)+1$ (2002, in Chinese, not held) with $f(k)\le R_k(3)-1$, proved in English in Eliahou's refereed paper (Integers 2020, Corollary 2) whose finite input is $R_4(3)\le62$. No source proving or refuting an exponential upper bound, and none proving a superexponential lower bound, was found in the search whose scope the Current assessment records; the values $f(k)$ are known exactly for $k\le5$ only, the last by Heule's 2017 computation. This is a bounded negative finding, not a certificate of openness.

Source. erdosproblems.com/483, accessed 2026-09-18: the problem page (labeled OPEN, with the site's note that no finite computation can settle it; last edited 1 September 2026; source keys [Er61, p. 233] and [Er65, p. 188]; OEIS A030126), its four-comment discussion thread (14 October 2025 to 23 April 2026) and its empty proof-claim tab. Cite as: T. F. Bloom, Erdős Problem #483, https://www.erdosproblems.com/483, accessed 2026-09-18.

References.

• [ACPPRT21] R. Ageron, P. Casteras, T. Pellerin, Y. Portella, A. Rimmel and J. Tomasik, New lower bounds for Schur and weak Schur numbers. arXiv:2112.03175 (v1 6 December 2021; v2 4 April 2022, read). Inequality (6) on p. 6, Corollary 2.9 on p. 7. Library home: ageron_2021_new_lower_bounds_schur_weak_schur.
• [Ex94] G. Exoo, A lower bound for Schur numbers and multicolor Ramsey numbers of $K_3$. Electron. J. Combin. 1 (1994), #R8, 3 pp. (DOI 10.37236/1188). Library home: exoo_1994_lower_bound_schur_numbers_multicolor_ramsey.
• [FrSw00] H. Fredricksen and M. M. Sweet, Symmetric sum-free partitions and lower bounds for Schur numbers. Electron. J. Combin. 7 (2000), #R32, 9 pp. (DOI 10.37236/1510). Library home: fredricksen_2000_symmetric_sum_free_partitions_lower_bounds.
• [He17] M. J. H. Heule, Schur Number Five. arXiv:1711.08076v1 (21 November 2017, the edition read); Proc. AAAI 32 (2018), DOI 10.1609/aaai.v32i1.12209 (not read); no file of either is held. Library home: heule_2017_schur_number_five.
• [El20] S. Eliahou, An adaptive upper bound on the Ramsey numbers $R(3,\ldots,3)$. Integers 20 (2020), Paper A54, 7 pp. (as the site's bibliography page gives it). Library home: eliahou_2020_adaptive_upper_bound_ramsey_numbers_r_3_3 (the journal's open-access file).
• [XXC02] X. Xu, Z. Xie and Z. Chen, Upper bounds for Ramsey numbers $R_n(3)$ and Schur numbers. Math. Econ. 19 (2002), 81--84. In Chinese; not held; no Crossref record; quoted second-hand from Eliahou (2020) and from the site.
• [AbHa72] H. Abbott and D. Hanson, A problem of Schur and its generalizations. Acta Arith. 20 (1972), no. 2, 175--187, DOI 10.4064/aa-20-2-175-187 (Crossref record). Not held; the thread's comment of 21 October 2025 cites its Corollary 2.1 for the inequality $\lim_{k\to\infty}S(k)^{1/k}\ge(2S(m)+1)^{1/m}$ in the Schur convention (equivalently $(2f(m)-1)^{1/m}$ with this page's $f=S+1$), which is quoted here second-hand from that comment.
• [Wa97] H. Wan, Upper bounds for Ramsey numbers $R(3,3,\cdots,3)$ and Schur numbers. J. Graph Theory 26 (1997), no. 3, 119--122, DOI 10.1002/(SICI)1097-0118(199711)26:3<119::AID-JGT1>3.0.CO;2-U (received 9 November 1990, revised 15 June 1993, per p. 119). Theorem 2.4 and Theorem 3.2, p. 121. The paper's Schur number $S_n$ is the least forcing number, this page's $f(n)$, with the values $S_1=2$, $S_2=5$, $S_3=14$, $S_4=45$ printed on p. 121. Library home: wan_1997_upper_bounds_ramsey_numbers_r_3_3_3_schur_numbers and its theorem_2_4 and theorem_3_2 pages.
• [Wh73] E. G. Whitehead, Jr., The Ramsey number $N(3,3,3,3;2)$. Discrete Math. 4 (1973), no. 4, 389--396, DOI 10.1016/0012-365X(73)90174-X. Not held; its bound $S(5)\le321$ is quoted second-hand from Exoo (1994), p. 1.
• [Sch16] I. Schur, Über die Kongruenz $x^m+y^m\equiv z^m\pmod p$. Jahresber. Deutsch. Math.-Verein. 25 (1916), 114--117; the Hilfssatz on p. 114. Library home: hilfssatz_p114.
• [Er61] P. Erdős, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. 6 (1961), 221--254; item 20, printed pp. 232--233. Library home: erdos_1961_unsolved_problems.
• [Er65] P. Erdős, Extremal problems in number theory. Proc. Sympos. Pure Math. VIII (1965), 181--189; printed p. 188. Library home: erdos_1965_extremal_problems_number_theory.
• Leads, not held: F. Rowley, An improved lower bound for $S(7)$ and some interesting templates, arXiv:2107.03560 (2021; its abstract states $S(7)\ge1696$ and $S(k+5)\ge376S(k)+160$); N. Bengone, A. Brouk, M. Grinsztajn, T. Helbert, B. Lugherini, A. Rimmel and J. Tomasik, Shifted S-templates and improved lower bounds for Schur numbers, arXiv:2607.15034v1 (16 July 2026; not filed; below); S. Hegde, A. Lott, G. Petridis and N. R. Ponagandla, Refined upper bounds on Schur-like numbers, arXiv:2608.03661 (August 2026; abstract only); R. W. Irving, An extension of Schur's theorem on sum-free partitions, Acta Arith. 25 (1973), 55--64 (the bound $S(k)\le\lfloor k!(e-1/24)\rfloor$, attested by Heule, p. 2).

Formalization. None: the directory FormalConjectures/ErdosProblems/ of formal-conjectures at the linked commit has no 483.lean, nor does main on 2026-10-07; the site page shows "Formalised statement? No"; the community database records the problem open (last changed 31 August 2025), not formalized, OEIS A030126 and no formal proof.

Current assessment

The question (site formulation of 2026-09-18). The statement above, labeled OPEN and marked by the site as beyond any finite computation, last edited 1 September 2026. The site's commentary, in this page's words: the values $f(k)$ are the Schur numbers; the best bounds for large $k$ are displayed as $(380)^{k/5}-O(1)\le f(k)\le(e-\tfrac16)k!$, with $380^{1/5}\approx3.2806$ noted; the lower bound is credited to Ageron, Casteras, Pellerin, Portella, Rimmel and Tomasik [ACPPRT21], who improved the earlier bounds of Exoo [Ex94] and of Fredricksen and Sweet [FrSw00]; the upper bound to Xu, Xie and Chen [XXC02], who improved those of Wan [Wa97] and Whitehead [Wh73], with Eliahou [El20] cited for a fuller account of it; the five known values of $f$, listed under Formulation, are given with OEIS A030126, the fifth credited to Heule [He17]; and Problem 183 is cross-referenced for the folklore bound $f(k)\le R(3;k)-1$. The thread, oldest first: a comment of 14 October 2025 by Zach Hunter observing that Schur never stated $f(k)\le R(3;k)-1$ as such, but that his argument is the standard proof of $R(3;k)\le ek!$ carried out without the language of graphs, so the bound is implicit in his paper; a comment of 21 October 2025 by Wouter CvB tracing the lower-bound base from Exoo's $(321)^{1/5}\approx3.1717$ ($321=2S(5)+1$) and Fredricksen and Sweet's $(1073)^{1/6}\approx3.1996$ ($1073=2S(6)+1$, via $S(6)\ge536$), both through the inequality $\lim_{k\to\infty}S(k)^{1/k}\ge(2S(m)+1)^{1/m}$ of Abbott and Hanson [AbHa72] (Corollary 2.1, as the comment cites it; equivalently $(2f(m)-1)^{1/m}$ in this page's convention), to Ageron et al.'s $(380)^{1/5}\approx3.2806$, which comes from their inequality (6) and not from that corollary; a comment of 7 November 2025 (the same account) tracing the upper-bound constant through $R(3;k)$ from Wan (1997), $\frac{e-e^{-1}+3}{2}k!+1$, to Xu, Xie and Chen (2002), $(e-\tfrac16)k!+1$, a paper written in Chinese, and pointing to Eliahou's English account, which adds bounds conditional on improved values for small $k$; and a comment of 23 April 2026 reporting that the reference [El19] failed to load (the page now cites [El20]). The site marks the first three as addressed. The proof-claim tab is empty.

The origins. Item 20 of Erdős's 1961 list, quoted under Formulation ([Er61], printed pp. 232--233), states Schur's theorem, defines $f(n)$ (one more than the site's $f$, as the two are worded; see Formulation), and records the conjecture $f(n)<c^n$ together with Turán's unpublished two-class result ($x+y=z$, $x\ne y$, forced in any two-class split of $n<k\le5n+3$ and not of $n<k\le5n+2$). The 1965 survey ([Er65], printed p. 188) poses the inverse question whether the least number $H(n)$ of sum-free classes of $\{1,\ldots,n\}$ exceeds $c\log n$, "very hard to decide", and notes Schur's $H(cn!)>n$. Schur's Hilfssatz (1916, p. 114, result page) is the origin of the factorial bound: every partition of $\{1,\ldots,N\}$ into $m$ classes with $N>m!e$ has a class containing $x$, $y$ and $y-x$, that is $f(m)\le\lfloor m!e\rfloor+1$ in the site's convention.

What is known: the bounds map. Exact values: $f(k)=2,5,14,45,161$ for $k=1,\ldots,5$. The first four are classical (Golomb and Baumert 1965 for $S(4)=44$, per Heule and Exoo); the fifth is Heule's main result $S(5)=160$ (2017; AAAI 2018), whose upper half is a two-petabyte unsatisfiability proof certified by a checker verified in ACL2, and whose lower half is Exoo's [[../library/ramsey_theory/exoo_1994_lower_bound_schur_numbers_multicolor_ramsey/lower_bound_p2|five-set partition of $[1,160]$]] (1994), recomputed here. Small values beyond: $f(6)\ge537$ and $f(7)\ge1681$ from Fredricksen and Sweet's constructions (2000; the 536 partition recomputed here), $f(7)\ge1697$ and $f(8)\ge5287$ from Rowley (2021; second-hand from his arXiv abstract and from Table 1 of Ageron et al.), and $f(9)\ge17\,804$, $f(10)\ge60\,949$, $f(11)\ge203\,829$, $f(12)\ge644\,629$ from Table 3 of Ageron et al.

Lower bound for all $k$: inequality (6), $S(n+5)\ge380S(n)+148$, from an explicit S-template found with a SAT solver, and its Corollary 2.9, growth rate at least $380^{1/5}\approx3.28$. The sources state the recursion and the growth rate, not the additive form "$(380)^{k/5}-O(1)\le f(k)$" that the site displays. The earlier bases $3.1717$ (Exoo) and $3.1996$ (Fredricksen and Sweet) are the thread's history; Rowley's abstract claims $3.273$ from $S(k+5)\ge376S(k)+160$, also superseded.

Upper bound for $k\ge4$: $f(k)\le(e-\tfrac16)k!$. The site's source is [XXC02], not held; the bound is proved in English as Eliahou's Corollary 2, $R_n(3)\le n!(e-1/6)+1$ for $n\ge4$, proved from his Theorem 1 (any bound $R_k(3)\le k!(e-q)+1$ with $k!q\in\mathbb N$ propagates through the Greenwood–Gleason recursion $R_n(3)\le n(R_{n-1}(3)-1)+2$) and the computational bound $R_4(3)\le62$ of Fettes, Kramer and Radziszowski (claims checked). With $S(n)\le R_n(3)-2$ (Eliahou's display (6); Fredricksen and Sweet's display (2); Exoo's $R_k(3)\ge S(k)+2$, all from the difference coloring of $K_{S(n)+1}$), $f(k)=S(k)+1\le R_k(3)-1\le(e-\tfrac16)k!$ for $k\ge4$. Eliahou's introduction (pp. 1--2) attests the chain Greenwood and Gleason $n!e+1$, Whitehead $n!(e-1/24)+1$, Wan $n!(e-e^{-1}+3)/2+1$, Xu–Xie–Chen $n!(e-1/6)+1$, and his Corollaries 3--4 show what the conjectured $R_4(3)=51$ (giving $e-5/8$) or $R_4(3)\le54$ (giving $e-1/2$) would yield; none of these is exponential. Wan's link of the chain is checked first-hand: his Theorem 2.4 (p. 121) proves $R_n(3)\le n!(e-e^{-1}+3)/2+1$ for $n\ge4$ from Folkman's $R_4(3)\le65$ by a parity refinement of the Greenwood–Gleason recursion, and his Theorem 3.2 (p. 121) gives $f(n)<n!(e-e^{-1}+3)/2-n+2$ for even $n\ge6$ directly in this page's convention; both are weaker than $(e-1/6)n!$ for every $n\ge4$, so they move no bound here. The folklore observation the site records, $f(k)\le R(3;k)-1$, is this $S(k)\le R_k(3)-2$; Problem 183's resolution ($\lim R(3;k)^{1/k}=\infty$) runs the wrong way for this problem, since a lower bound on $R_k(3)$ says nothing about $f(k)$.

So the question "is $f(k)<c^k$" is open with a gap between exponential and factorial growth: no source gives an upper bound of the shape $c^k$, and no source gives a lower bound growing faster than an exponential. The inequality the thread cites, $\lim S(k)^{1/k}\ge(2S(m)+1)^{1/m}$, that is $\lim f(k)^{1/k}\ge(2f(m)-1)^{1/m}$ ([AbHa72], Corollary 2.1 as the comment cites it; the paper is not held), presupposes that the limit exists, and Eliahou's Section 3.1 reports that $\lim R_n(3)^{1/n}$ is "known by [2] to exist" (Chung and Grinstead 1983; not held); granted the existence, the exponential question is whether $\lim f(k)^{1/k}$ is finite.

Forum and AI-assisted items (leads with provenance, not status). The arXiv preprint of Bengone, Brouk, Grinsztajn, Helbert, Lugherini, Rimmel and Tomasik (arXiv:2607.15034v1, 16 July 2026, four pages) reports a "shifted S-template" of width 10 giving the recurrence $S(k+2)\ge10S(k)+2$ (Theorem 2, a finite template check), hence $S(8)\ge5\,362$ and $S(13)\ge2\,038\,282$, improving the listed $5\,286$ and $2\,011\,290$. Its abstract declares, and its conclusion repeats, that the construction "was discovered during a conversation with ChatGPT 5.5 Pro" and was then "refined, verified, and extended" by the authors. It is unrefereed, the site's page (last edited 1 September 2026) does not cite it, and its asymptotic content ($10^{1/2}\approx3.16<380^{1/5}$) does not move the growth-rate bound; it is recorded as a lead for the small values $k=8$ and $k=13$ only and is not filed. Hegde, Lott, Petridis and Ponagandla (arXiv:2608.03661, August 2026, abstract only) bound Schur-like numbers for $x_1+\cdots+x_{m+1}=y_1+\cdots+y_m$ by $3^r(r!)^{1/m}$; for $m=1$ that is weaker than $(e-1/6)r!$ and does not bear on the bounds here. No proof claim exists on the site.

Search scope. None of the routes below found an exponential upper bound, a superexponential lower bound, a new exact value, or a proof claim.

• The site: problem page, discussion thread and proof-claim tab; the formal-conjectures directory listing at the commit linked under Formalization (no file); the community database at the commit linked under Formalization; the site's bibliography page for [El20].
• arXiv: the abstract pages of 2112.03175 (two versions, no journal reference), 1711.08076 (one version, "accepted by AAAI 2018") and 2607.15034 (one version); the API queries abs:"Schur number" by date (17 records, the newest the Bengone preprint, then modular and off-diagonal Schur numbers; none on the growth of $S(k)$ beyond the preprint) and abs:"sum-free partition" (one record, on weak Schur partitions); the API records of 2608.03661 and 2107.03560 (abstracts as stated above). The API searches titles and abstracts only, so its zeros are weak.
• Crossref: the records of Exoo (DOI 10.37236/1188), Fredricksen–Sweet, Wan, Whitehead (DOI 10.1016/0012-365X(73)90174-X), Irving 1973 and Heule (AAAI 2018); bibliographic queries for Xu–Xie–Chen, Eliahou, Ageron et al. and Bengone et al. (no records: the first two journals carry no DOIs, the last two are preprints).
• Semantic Scholar: the nine papers citing Ageron et al., scanned by title (the Bengone preprint, the Schur-like upper bounds, Gallai–Schur papers, odd-cycle Ramsey bounds; none an exponential upper bound); the citation list of Heule 2017 returned no records.
• OEIS A030126 (2, 5, 14, 45, 161 in the site's convention; a comment "a(6) ≥ 537, a(7) ≥ 1681"; nothing beyond the sources above).
• The primary sources: Heule pp. 1--2, Exoo pp. 1--3, Fredricksen–Sweet pp. 1--3 and 6--8, Ageron et al. pp. 1--3 and 5--7, Eliahou pp. 1--7, Schur p. 114, Erdős 1961 pp. 232--233, Erdős 1965 p. 188, and the Bengone preprint pp. 1--4.

Not searched: MathSciNet, zbMATH, Google Scholar, X. Not held: [XXC02], [Wh73], Irving 1973, [AbHa72], Rowley 2021 (abstract only), Turán's unpublished result, the AAAI version of [He17]. [Wa97] lies outside the search.

Remaining gaps. (1) The exponential question is untouched: the gap between $c\cdot3.28^k$ and $(e-1/6)k!$ is the whole problem; both shapes, exponential below and factorial above, go back to Schur 1916 (the Hilfssatz on p. 114 and the construction $N_{m+1}\ge3N_m+1$ on pp. 116--117, hence $S(m)\ge(3^m-1)/2$), and later work moved only the constants. (2) The site's upper-bound sources [XXC02] and [Wh73] are not held; [Wa97] gives only the weaker $n!(e-e^{-1}+3)/2+1$; the bound is proved in Eliahou's Corollary 2, whose finite input $R_4(3)\le62$ is a computational theorem taken at statement level. (3) The site's additive display of the lower bound, $(380)^{k/5}-O(1)\le f(k)$, is not the form the sources state; Ageron et al. give the recursion (6) and the growth rate of Corollary 2.9. (4) The Bengone et al. records for $k=8$ and $13$ are an unrefereed, AI-assisted lead. (5) The records $S(7)\ge1696$ and $S(8)\ge5286$ (Rowley) are second-hand. (6) The site cites p. 233 of [Er61]; the item begins on p. 232. (7) Ageron et al. is read as an arXiv preprint (v2), and Heule as the arXiv v1 rather than the AAAI version; neither file is held.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1965_extremal_problems_number_theory
• erdos_1961_unsolved_problems
• ageron_2021_new_lower_bounds_schur_weak_schur
• ageron_2021_new_lower_bounds_schur_weak_schur / corollary_2_9
• ageron_2021_new_lower_bounds_schur_weak_schur / inequality_6
• ageron_2021_new_lower_bounds_schur_weak_schur / theorem_2_3
• ageron_2021_new_lower_bounds_schur_weak_schur / theorem_3_1
• ageron_2021_new_lower_bounds_schur_weak_schur / theorem_3_17
• eliahou_2020_adaptive_upper_bound_ramsey_numbers_r_3_3
• eliahou_2020_adaptive_upper_bound_ramsey_numbers_r_3_3 / corollary_2
• eliahou_2020_adaptive_upper_bound_ramsey_numbers_r_3_3 / theorem_1
• exoo_1994_lower_bound_schur_numbers_multicolor_ramsey
• exoo_1994_lower_bound_schur_numbers_multicolor_ramsey / lower_bound_p2
• fredricksen_2000_symmetric_sum_free_partitions_lower_bounds
• fredricksen_2000_symmetric_sum_free_partitions_lower_bounds / constructions_p6
• heule_2017_schur_number_five
• heule_2017_schur_number_five / extreme_certificates_p7
• heule_2017_schur_number_five / main_result
• schur_1916_uber_die_kongruenz
• schur_1916_uber_die_kongruenz / hilfssatz_p114
• schur_1916_uber_die_kongruenz / lower_bound_p117
• wan_1997_upper_bounds_ramsey_numbers_r_3_3_3_schur_numbers
• wan_1997_upper_bounds_ramsey_numbers_r_3_3_3_schur_numbers / theorem_2_4
• wan_1997_upper_bounds_ramsey_numbers_r_3_3_3_schur_numbers / theorem_3_2