Pith Number

pith:ACG7FNTJ

pith:2025:ACG7FNTJH63SUBMYPQZYKCR6K4

not attested not anchored not stored refs pending

An Erd\H{o}s--Szekeres type result for words with repeats

Abigail Ollson, Jun Yan, Kyle Celano, Niraj Velankar

Every word with kn^6+1 repeats must contain one of seven specific patterns.

arxiv:2510.23573 v4 · 2025-10-27 · math.CO

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\usepackage{pith}
\pithnumber{ACG7FNTJH63SUBMYPQZYKCR6K4}

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Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

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

We show that every word with kn^6+1 repeats contains one of the following patterns: 0^{k+2}, 0011⋯nn, nn⋯1100, 012⋯n012⋯n, 012⋯nn⋯210, n⋯210012⋯n, n⋯210n⋯210. Moreover, when k=1, this is best possible by constructing a word with n^6 repeats that does not contain any of these patterns.

C2weakest assumption

The central claim rests on the specific choice of the seven target patterns as the complete set of unavoidable configurations once the repeat count exceeds the stated threshold; if a different or larger set of patterns were required to be avoided, the quantitative bound would not necessarily hold.

C3one line summary

Every word with kn^6+1 repeats contains one of the patterns 0^{k+2}, 0011⋯nn, nn⋯1100, 012⋯n012⋯n, 012⋯nn⋯210, n⋯210012⋯n, or n⋯210n⋯210, with the bound tight for k=1 via an explicit construction.

Formal links

1 machine-checked theorem link

Receipt and verification

First computed	2026-05-20T02:05:37.511543Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

008df2b6693fb72a05987c33850a3e572080bc6df2eeebafd4e5364bd836f269

Aliases

arxiv: 2510.23573 · arxiv_version: 2510.23573v4 · doi: 10.48550/arxiv.2510.23573 · pith_short_12: ACG7FNTJH63S · pith_short_16: ACG7FNTJH63SUBMY · pith_short_8: ACG7FNTJ

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/ACG7FNTJH63SUBMYPQZYKCR6K4 \
  | 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: 008df2b6693fb72a05987c33850a3e572080bc6df2eeebafd4e5364bd836f269

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "c9b591c8cda0dc28b2a84936a36a30821fd296c8adc04e928ebb60340e0664b7",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2025-10-27T17:42:16Z",
    "title_canon_sha256": "3bda29e12e3072a3a47d471410bb5d78e680b8d8cdbcc19cc18f4e38d9724b52"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2510.23573",
    "kind": "arxiv",
    "version": 4
  }
}