Erdős problem #483 — verified partial progress

Date of live-page check and literature cutoff: 2026-07-26 UTC.

Claim labels used below:

• (a) elementary-rigorous: proved here from definitions, with finite arithmetic rechecked by the standalone script where applicable.
• (b) rigorous-modulo-named-theorem/source: depends on the cited theorem or on faithful transcription from a primary source.
• (c) plausible/structural-unverified: an explicitly identified heuristic or estimate, not used as a theorem.
• (d) computational-only: an exhaustive finite computation whose code and output are supplied, but not a separately formalized proof.

0. Mandatory live-page gate

I fetched the live problem page and its discussion thread through the Bright Data browser on 2026-07-26. (b)

The live statement, verbatim, is:

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 \(c>0\)?

The gate was clear:

• status: OPEN;
• claimed proofs: 0;
• “Currently working on this problem”: None;
• “Interested in collaborating”: None;
• page last edited: 10 April 2026. (b)

The page states

and \(f(1),\ldots,f(5)=2,5,14,45,161\). It attributes the lower bound to Ageron–Casteras–Pellerin–Portella–Rimmel–Tomasik and the upper bound to Xu–Xie–Chen, with Eliahou as an accessible English account of the latter. (b)

I also read all four comments:

1. Alfaiz, 23 April 2026: the site could not load its [El19] reference.
2. Wouter CvB, 7 November 2025: supplied the Wan and Xu–Xie–Chen

improvements to the multicolour-triangle Ramsey upper bound and pointed to Eliahou’s English proof; the site says it was updated.

1. Wouter CvB, 21 October 2025: supplied the Fredricksen–Sweet and

Ageron et al. lower-bound history and the Abbott–Hanson product inequality; the site says it was updated.

1. zach hunter, 14 October 2025: corrected the attribution of

\(f(k)\leq R(3;k)-1\) from an explicit Schur observation to a folklore or implicit observation; the site says it was updated. (b)

None is a claimed proof or a current-worker marker, so the requested stop condition does not apply.

1. Notation and verified literature state

It is convenient to use the other standard convention

so \(S(k)\) is the largest \(n\) for which \([1,n]\) has a \(k\)-coloring with no monochromatic \(a+b=c\). (a)

Primary sources actually checked:

• Erdős’s 1961 paper, Some unsolved problems,

p. 233, defines the same threshold and records the conjectures \(f(n)<c^n\) and \(f(n)^{1/n}\to C\). (b)

• Heule, Schur Number Five, proves

\(S(5)=160\) by SAT, with a formally checked certificate; the paper reports a multi-CPU-year computation and a two-petabyte proof. (b)

• Fredricksen–Sweet,

Symmetric Sum-Free Partitions and Lower Bounds for Schur Numbers, prints an explicit six-coloring of \([1,536]\). (b)

• Ageron et al.,

arXiv:2112.03175v2, define \(S\)-templates, print width-111 and width-380 templates, and derive \[ S(k+4)\geq111S(k)+43,\qquad S(k+5)\geq380S(k)+148. \] (b)

• Eliahou,

arXiv:1912.05353, gives an English proof that \(R_k(3)\leq (e-\tfrac16)k!+1\) for \(k\geq4\), hence the live-page upper bound through \(f(k)\leq R_k(3)-1\). (b)

• A literature search found a relevant paper posted only ten days before

this run, so it is not yet reflected on the live page: Bengone–Brouk–Grinsztajn–Helbert–Lugherini–Rimmel–Tomasik, Shifted S-templates and improved lower bounds for Schur numbers, arXiv:2607.15034v1, submitted 16 July 2026. It proves \[ S(k+2)\geq10S(k)+2 \] and reports \(S(8)\geq5362\) and \(S(13)\geq2\,038\,282\). (b)

I verified that all five arXiv identifiers above exist and that their abstracts/PDFs contain the claims just attributed to them. I could not obtain the full 2002 Chinese Xu–Xie–Chen paper; I therefore use Eliahou’s explicit reproduction as the named theorem for the upper bound, rather than pretending to have checked the Chinese proof. Searches through 2026-07-26 found no paper claiming the requested exponential upper bound. That last statement is a search report, not a proof that no obscure source exists. (b)

2. Explicit construction reverified from scratch

2.1 The shifted template

The new width-10 template uses two new colors \(A,B\). If the base sum-free coloring is \(g:[1,p]\to[k]\), write the ten positions in row \(\alpha\) as follows:

Append the two-cell tail \(A,B\). (b)

Here is a short independent proof of the recurrence. In a sum inside the main block, if the column addition has carry \(\epsilon\in\{0,1\}\), the target row is \(\gamma=\alpha+\beta-1+\epsilon\). A base-color cell in a row has reference shift \(d=0\), except column 1 of a non-first row, which has \(d=-1\). Direct enumeration of the six possible first/middle row transitions and the \(10^2\) column pairs shows that whenever all three cells are base-color cells,

Consequently

so equality of all three inherited colors would contradict sum-freeness of \(g\). The same enumeration finds no triple of identical ordinary labels. For the tail, possible \(A\)-columns have residues \(\{2,6,7\}\pmod {10}\), no two of which sum to the first tail residue \(1\); possible \(B\)-columns have residues \(\{1,2,5,8\}\), and the only pair summing to the second tail residue \(2\) is \(1+1\). The label \(B\) occurs in residue 1 only at the literal integer 1, whose self-sum is 2, not \(10p+2\). Thus the tail is also safe. This proves, for every sum-free base coloring of length \(p\geq3\),

This proof is independent of the SAT search that discovered the template. The checker enumerates all 300 symbolic transition/column cases. (a)

2.2 A concrete eight-coloring

The standalone script embeds the half-partition printed by Fredricksen–Sweet, expands its symmetry \(x\leftrightarrow537-x\) (respecting their exceptional pair \(179,358\)), and first checks every candidate sum in the resulting six-coloring of \([1,536]\). It then applies the table above and appends \(A,B\). A second validator, implemented independently with integer bitsets, checks the same two colorings. (d)

The result is an explicit eight-coloring of

with class sizes

The independent pair loop checked all \(7\,187\,761\) pairs \(1\leq a\leq b\), \(a+b\leq5362\), and found no monochromatic solution. The deterministic SHA-256 of the one-byte-per-entry color sequence (colors encoded \(1,\ldots,8\)) is

11ac22bfc0ce56fd3d4055fb74207abdeb2ba7734a17ece7ebea4355b23b1815

Therefore

This numerical bound is already announced in arXiv:2607.15034v1; the progress here is a self-contained reconstruction and exhaustive checker, not a priority claim for the number. (d)

3. An infinite derived subsequence

The width-380 Ageron template has five ordinary colors, special-color minimum \(149\), and gives

The standalone script transcribes all 380 entries and checks from the definition that each class is sum-free and that the five ordinary classes are also sum-free modulo 380. The usual row-expansion argument then proves (3): ordinary-color sums are excluded modulo the width, and a wrapped sum of three special cells projects to a forbidden sum in the base coloring. (a)

Starting from \(S(6)\geq536\), apply (3) \(t\) times and then (1) once. This gives, for every integer \(t\geq0\),

or, in the live page’s convention,

Divisibility and all table entries below are recomputed with integer arithmetic in the checker. (a)

The 2026 paper prints the first two values; (4) is the immediate infinite composition with the width-380 construction. I do not claim that every later entry is a literature-wide record, only that each displayed bound follows rigorously from the two verified templates. (a)

This does not improve the asymptotic base on the live page:

Thus (4) is finite/infinite-subsequence progress, not progress toward an exponential upper bound. (a)

4. Two exact finite diagnostics on the best older templates

4.1 Repair of the printed width-111 certificate

Appendix A, Table 6 of arXiv:2112.03175v2 purports to partition \([1,111]\), but its printed rows cover only 110 integers: 49 is absent. Adding 49 to color 1 repairs the table:

color 1 =
1,5,8,12,14,21,23,30,32,36,39,43,45,49,52,103,106,110.

Exhaustive checking shows that adding 49 to color 4 also gives an \(S\)-template, but only color 1 preserves the stated symmetry. Indeed, the symmetry reflection pairs 49 with 106, which is in color 1. The repaired assignment has SHA-256

7dc3bd4a839fc1c2805af7b65e97c9b6f9e8558ac687375f8a28110910d3adff

and the checker verifies every ordinary and wrapped sum. This is a correction to the printed explicit certificate; it does not change the coefficient 111 or close the main problem. (d)

4.2 The width-380 last-row constant is exactly blocked at 149

For the published width-380 template, the standard tail of length 148 passes both conditions of Ageron et al.’s Proposition 2.5. At the next tail position, residue 149 is forbidden in every color by these same-color pairs:

Hence no color can be assigned to tail cell 149 under that proposition. The additive constant 148 is therefore exactly maximal for this fixed template and this last-row extension mechanism. This does not rule out a different width-380 template or a more general shifted construction. (a)

5. Exact reduction of what the main question still needs

Let \(\chi_{\rm Schur}(N)\) be the least number of colors needed to color \([1,N]\) with no monochromatic \(a+b=c\). Equivalently, it is the chromatic number of the constraint system containing the triples \(\{a,b,a+b\}\) for \(a<b\) and the two-vertex constraints \(\{a,2a\}\). Then

Inverting (6) shows:

Additive changes in the exponent or threshold do not affect this equivalence. (a)

The live-page factorial upper bound and the template lower bound translate to

The left inequality follows by choosing the largest \(k\) with \((e-\tfrac16)k!\leq N\) and using Stirling; the right follows by inverting the template construction. (a), using the source bounds (b)

Thus the exact missing direct lemma is:

Prove a uniform lower bound \(\chi_{\rm Schur}(N)\geq\delta\log N\) for some \(\delta>0\) and all sufficiently large \(N\).

Current machinery supplies only \(\delta\asymp1/\log\log N\). On the Ramsey route, the analogous missing ingredient is a bounded-ratio estimate \(R_{k+1}(3)\leq C R_k(3)\), or another argument of comparable uniform strength. Eliahou explicitly notes that even finiteness of \(\lim R_k(3)^{1/k}\) is unknown. Improving any fixed small Ramsey number in his recursion changes only the constant multiplying \(k!\); it cannot remove the factorial. (a), with Eliahou’s recurrence (b)

This also explains why no finite table, including (4), can close the problem: the required step is uniform in \(k\) (or \(N\)), not one more finite coloring.

6. Computation, reproducibility, and the honest wall

Standalone verifier:

runs/erdos483_wave5c_verify.py

Run:

python runs/erdos483_wave5c_verify.py

The core exhaustive check is deliberately simple:

for a in range(1, n // 2 + 1):
    for b in range(a, n - a + 1):
        assert not (
            color[a] == color[b] == color[a + b]
        )

The shifted construction itself is generated from the two ten-symbol rows, not stored as a claimed list. The complete primary-source base partition and both older templates are embedded in the script, which checks disjointness and coverage before checking sums.

Observed clean-run output:

printed width-111 missing entry: 49
valid width-111 repairs for 49: [1, 4]
unique symmetry-preserving repair: color 1
width-380 Proposition-2.5 tail maximum: 148
shifted-template symbolic transitions checked: 300
base [1,536] triples checked: 71824
base [1,536] independent bitset shifts: 536
constructed [1,5362] triples checked: 7187761
constructed [1,5362] independent bitset shifts: 5362
constructed [1,5362] sha256: 11ac22bfc0ce56fd3d4055fb74207abdeb2ba7734a17ece7ebea4355b23b1815
ALL CHECKS PASSED

On this VM the final full run took 0.63 seconds and about 17 MB RSS. (d)

I also tried the first unrestricted five-color ordinary \(S\)-template instance beyond width 111. A modern SAT solver did not decide width 112 within the imposed 120-second exploratory cap, so no claim is based on that run. The 2026 shifted-template paper reports a 20-hour timeout already for the much smaller \(r=3,q=34,\Delta=\{-3,\ldots,3\}\) instance and estimates encoding size as \(\sim q^2|\Delta|^3\). At fixed shift range, merely scaling clause-touch work from \(q=34\) gives factors about 10.8 for \(q=112\) and 125.6 for \(q=381\). If the reported run was single-core, this suggests optimistic linear-work floors of roughly 220 and 2,500 core-hours per configuration; SAT search is nonlinear, so these are budgeting floors, not credible completion forecasts. (c)

More importantly, even a successful larger-template search improves a lower bound and cannot establish (7). The mathematical wall is the missing logarithmic chromatic lower bound, not a few more CPU-hours.

PARTIAL: Explicitly reverified an 8-coloring of [1,5362], derived a checked infinite lower-bound subsequence, repaired the printed width-111 template, and proved the width-380 tail constant 148 locally maximal; the exponential upper bound remains exactly the missing \(\chi_{\rm Schur}(N)=\Omega(\log N)\) lemma.