Pith Number

pith:U5ENM3MI

pith:2026:U5ENM3MIVCKQGD7TWSBQSLVTT5

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Improved Ramsey bounds for generalized Schur equations

Cosmin Pohoata, Eion Mulrenin, Michael Zheng, Rafael Miyazaki

For N larger than (2m+1)^r times (r!)^{1/m}, any r-coloring of [N] forces a monochromatic solution to x1+...+x_{m+1}=y1+...+ym.

arxiv:2605.15147 v1 · 2026-05-14 · math.CO · math.NT

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\pithnumber{U5ENM3MIVCKQGD7TWSBQSLVTT5}

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Claims

C1strongest claim

for m, r ∈ ℕ and N > (2m+1)^r (r!)^{1/m}, every r-coloring of the integers in the interval [N] contains a monochromatic solution to the equation x1 + ⋯ + x_{m+1} = y1 + ⋯ + ym

C2weakest assumption

The derivation of the explicit bound from standard combinatorial tools (pigeonhole or iterative coloring arguments) holds without hidden dependencies on m or r that would invalidate the inequality for large values.

C3one line summary

Improved explicit upper bounds on the Ramsey numbers guaranteeing monochromatic solutions to x1+...+x_{m+1}=y1+...+ym in r-colorings of [N], with the bound N>(2m+1)^r (r!)^{1/m} and optimality of N=2^r for some m.

References

22 extracted · 22 resolved · 0 Pith anchors

[1] H.AbbottandD.Hanson,A problem of Schur and its generalizations, ActaArith.20(1972), 175–187; MR0319934 2 1972

[2] R. Ageron, P. Casteras, T. Pellerin, Y. Portella, A. Rimmel, J. Tomasik,New lower bounds for Schur and weak Schur numbers(2022), preprint available athttps://arxiv.org/abs/2112.031752 2022

[3] M. Axenovich, W. Cames von Batenburg, O. Janzer, L. Michel, and M. RundströmAn improved upper bound for the multicolor Ramsey number of odd cycles(2025), preprint available athttps://arxiv.org/abs/251 2025

[4] P. Balister, B. Bollobás, R. Morris, J. Sahasrabudhe, and M. Tiba,Covering intervals with arithmetic progressions, Acta Math. Hungar.161(2020), 197–200; MR4110365 7 2020

[5] R. B. Crittenden and C. L. Vanden Eynden,A proof of a conjecture of Erdős, Bull. Amer. Math. Soc.75(1969), 1326–1329; MR0249351 3, 7 1969

Receipt and verification

First computed	2026-05-17T21:40:25.506232Z
Last reissued	2026-05-17T21:57:18.826133Z
Builder	pith-number-builder-2026-05-17-v1
Signature	unsigned_v0
Schema	pith-number/v1.0

Canonical hash

a748d66d88a895030ff3b483092eb39f42f1eb297a68ef42cb433040ca2a7f6f

Aliases

arxiv: 2605.15147 · arxiv_version: 2605.15147v1 · pith_short_12: U5ENM3MIVCKQ · pith_short_16: U5ENM3MIVCKQGD7T · pith_short_8: U5ENM3MI

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/U5ENM3MIVCKQGD7TWSBQSLVTT5 \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: a748d66d88a895030ff3b483092eb39f42f1eb297a68ef42cb433040ca2a7f6f

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "7ddfed18f5a04c8e9c3d5f20bac5da50f42db08844541595b464a2d907f4d450",
    "cross_cats_sorted": [
      "math.NT"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-05-14T17:49:42Z",
    "title_canon_sha256": "fa0e689de7314aaa9a400c21b9b81c228aa37e5e3372ba8b61ed8dc054384368"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.15147",
    "kind": "arxiv",
    "version": 1
  }
}