Problem 483

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 $c>0$ ?

Source: erdosproblems.com/483

Number theory
Additive combinatorics
Ramsey theory

Wiki page

Status

Open

No claim settles this problem.

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.

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 et al. 2021.
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.
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 and Sweet 2000.
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.
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 (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 and its Wan 1997, Theorem 2.4 and Wan 1997, 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: Schur 1916, Hilfssatz, p. 114.
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: Erdős 1961.
Er65: P. Erdős, Extremal problems in number theory. Proc. Sympos. Pure Math. VIII (1965), 181--189; printed p. 188. Library home: Erdos 1965. - 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).

Problem 483

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Proof claims

Discussion