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A new lower bound for the sixth Schur number: $S(6)\ge 537$
Zhiyang Chen, Ligeng Zhu, Fan Long
TL;DR
The record lower bound for S(6) had been S(6) ≥536 since 2000, and this paper improves it by one. The authors give an explicit partition of {1, . . . , 537} into six sum-free sets and apply known recursive constructions to derive other bounds. S(6) ≥ 537, with derived bounds S(8) ≥ 5,372, S(11) ≥ 204,208, S(13) ≥ 2,042,082, and R6(3) ≥ 539.
Problem
The record lower bound for S(6) had been S(6) ≥536 since 2000, and this paper improves it by one.
Source: §1 · p. 1Method
Results
Takeaways & Limitations
Abstract
from arXiv · showhide
We give an explicit partition of $\{1,\ldots,537\}$ into six sum-free sets, establishing $S(6)\ge 537$. This improves the lower bound 536 of Fredricksen and Sweet (2000). Known recursive constructions then give the improved lower bounds $S(8)\ge 5\,372$, $S(11)\ge 204\,208$, $S(13)\ge 2\,042\,082$, and $R_6(3)\ge 539$. We also report 131 pairwise inequivalent partitions found by the search. We describe their structure and provide the complete partitions in the supplementary material. These partitions provide starting points for future searches and structural studies of Schur partitions.
1 Introduction
The paper improves the long-standing lower bound for the sixth Schur number by one, giving an explicit partition into six sum-free sets. This single partition establishes S(6) ≥ 537.
1 Introduction
- S(6) ≥ 537, improving the ≥536 bound that had stood since 2000.Fredricksen and Sweet found the earlier bound by searching symmetric sum-free partitions.Source: §1 · p. 1
- The six listed classes partition {1, . . . , 537} and are verified to contain no x, y, and x + y within any class.The verification checks all within-class sums, including x = y; the complete element lists are supplied in the paper.Source: §1 · pp. 1–2
- The theorem requires only one such partition; the paper also reports 131 partitions found by search and provides them in supplementary material.The supplementary material contains the complete partitions.Source: §1 · p. 2
2 Consequences for other bounds
Classical recurrences and template constructions propagate the improved S(6) bound to larger Schur and Ramsey numbers. The paper also explicitly constructs the eight-color partition and checks four resulting bounds against previously published values.
2 Consequences for other bounds
- The reported recurrences are S(k + 2) ≥ 10S(k) + 2 [5] and S(k + 5) ≥ 380S(k) + 148 [4], giving two explicit lower-bound relationships in k.Source: §2 · p. 2
- The Ramsey consequence uses R_k(3) ≥ S(k) + 2, obtained by coloring edges of K_{n+1} by the class of the endpoint-distance.A sum-free k-partition of [1, n] gives an edge coloring without monochromatic triangles.Source: §2 · p. 2
- S(8) ≥ 5,372, S(11) ≥ 204,208, S(13) ≥ 2,042,082, and R6(3) ≥ 539 follow from the improved S(6) bound.The S(8) bound is also realized by an explicit partition into eight sum-free sets.Source: §2 · p. 3
- The paper checks four cases in which the recurrence-derived values change previously published bounds; other larger-k bounds are not listed individually.The six- and eight-color bounds are supported by explicit partitions.Source: §2 · p. 3, Table 1 · p. 3
3 The 537-partitions and their families
The paper organizes 131 retained partitions into five families and describes structural features of the example partition. The retained sample is not an exhaustive classification of all six-color partitions.
3 The 537-partitions and their families
- The example is palindromic, has class minima 1, 2, 3, 4, 38, and 151, and its sixth class contains [237, 301].Its classes are even sum-free modulo 538.Source: §3 · p. 3
- In frame g, color 6 is absent below 151 and fills the interval's middle, while color 5 is absent below 38 and just below 151.The frame uses Q = 151 and d = 38.Source: §3 · p. 3
- The example's first class is C*, formed by taking positions whose folded values f(x) lie in B, a set of 38 numbers.C* has 76 numbers; B is specified by the construction in the text.Source: §3 · p. 3
- The 131 listed partitions are pairwise inequivalent under multiplication by units modulo 538 followed by color renaming, and are grouped into five families.Every class is sum-free modulo 538; partitions in F1–F4 agree with the example in at least 462 of 537 positions.Source: §3 · p. 3
- The families describe the retained sample, not every possible partition of {1, . . . , 537} into six sum-free sets.Table 2 reports observed intervals within families, and its h quantity counts folded values rather than individual positions.Source: §3 · p. 4, Table 2 · p. 4
4 Construction of the families
The families vary reflection symmetry and frame conditions while enforcing sum-freeness modulo 538. The 131 retained partitions describe structural variations, not an exhaustive classification.
4 Construction of the families
- F1 and F2 impose reflection symmetry, sum-freeness modulo 538, and frame g, differing in whether the first class is C* or differs by one reflected pair.Source: §4 · p. 4
- F3 retains the frame conditions but removes reflection symmetry, while F4 retains reflection symmetry but removes the solid-middle condition.Source: §4 · p. 4
- F5 starts from the Theorem 1 coloring c, renames each c_m to maximize agreement with c, and imposes sum-freeness modulo 538 without reflection symmetry or the solid-middle condition.Source: §4 · p. 4
- Theorem 1’s example suffices for the lower bound, while the other partitions are structural variations rather than an exhaustive classification of six-color partitions of [1, 537].Source: §4 · p. 4