Pith Number

pith:WBSKRL5Y

pith:2026:WBSKRL5YBGV6F5OJSD3DA4N3HD

not attested not anchored not stored refs resolved

Monochromatic unit equilateral triangle on low-dimensional spheres

Gennian Ge, Xiaochen Zhao

There exists a 2-coloring of the 2-sphere of radius 1/√2 with no monochromatic unit equilateral triangle, but every 2-coloring of the 3-sphere contains one.

arxiv:2605.16958 v1 · 2026-05-16 · math.CO · math.MG

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Record completeness

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3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

there exists a 2-coloring of S^2(1/√2) with no monochromatic unit equilateral triangle, whereas every 2-coloring of S^3(1/√2) contains one.

C2weakest assumption

The unit equilateral triangle is defined via Euclidean chord distances in the ambient R^{n+1}, and the sphere radius 1/√2 is chosen so that the target distance 1 corresponds to a geometrically natural configuration on that sphere.

C3one line summary

The exact threshold dimension for monochromatic unit equilateral triangles in 2-colored spheres of radius 1/√2 is 3.

References

16 extracted · 16 resolved · 0 Pith anchors

[1] D. Cherkashin and V. Voronov. On the chromatic number of 2-dimensional spheres.Discrete Comput. Geom., 71(2):467–479, 2024 2024

[2] G. Currier, K. Moore, and C. H. Yip. Any two-coloring of the plane contains monochromatic 3-term arithmetic progressions.Combinatorica, 44(6):1367–1380, 2024 2024

[3] P. Erd˝ os, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer, and E. G. Straus. Euclidean Ramsey theorems. I.J. Combinatorial Theory Ser. A, 14:341–363, 1973 1973

[4] P. Erd˝ os, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer, and E. G. Straus. Euclidean Ramsey theorems. II. In A. Hajnal, R. Rado, and V. T. S´ os, editors,Infinite and Finite Sets, volume 1975

[5] P. Erd˝ os, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer, and E. G. Straus. Euclidean Ramsey theorems. III. In A. Hajnal, R. Rado, and V. T. S´ os, editors,Infinite and Finite Sets, volum 1975

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:03:32.881614Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

b064a8afb809abe2f5c990f63071bb38f037853ee50decdf72ff9619ffea8d4b

Aliases

arxiv: 2605.16958 · arxiv_version: 2605.16958v1 · doi: 10.48550/arxiv.2605.16958 · pith_short_12: WBSKRL5YBGV6 · pith_short_16: WBSKRL5YBGV6F5OJ · pith_short_8: WBSKRL5Y

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/WBSKRL5YBGV6F5OJSD3DA4N3HD \
  | 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: b064a8afb809abe2f5c990f63071bb38f037853ee50decdf72ff9619ffea8d4b

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "9063d664ad483d3d0d442f7c996bf95e0c9073c3cfe8770b8d7e163954cc100a",
    "cross_cats_sorted": [
      "math.MG"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-05-16T12:21:48Z",
    "title_canon_sha256": "a9e139fa315b61772b69594abb87e3e1c32aac32f27c2b4f72f5fecbfcf957c1"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.16958",
    "kind": "arxiv",
    "version": 1
  }
}