Pith Number

pith:WHGUBREO

pith:2025:WHGUBREODXLU5BOUGEMTKKKAUF

not attested not anchored not stored refs resolved

Some questions on entangled linear orders

Lorenzo Notaro, Maxwell Levine, Rapha\"el Carroy

Under the continuum hypothesis there exist n-entangled linear orders that are not n+1-entangled for every n.

arxiv:2507.17503 v2 · 2025-07-23 · math.LO · math.CO

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{WHGUBREODXLU5BOUGEMTKKKAUF}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

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

If CH holds, then for every n > 0 there is an n-entangled linear order which is not (n+1)-entangled; similar conditional existence statements hold for homeomorphic reals with differing entanglement, an entangled Pi1^1 set under V=L, and a 2-entangled non-separable order under diamond.

C2weakest assumption

The background definitions of n-entangled linear order and of the relevant descriptive-set-theoretic complexity classes (Pi1^1) as introduced by Abraham-Shelah and used in the constructions, together with the assumption that the ambient universe satisfies the stated extra axioms (CH, V=L, diamond).

C3one line summary

Under CH the authors construct n-entangled linear orders that fail to be (n+1)-entangled, plus homeomorphic reals with differing entanglement, an entangled Pi1^1 set under V=L, and a 2-entangled non-separable order under diamond.

References

9 extracted · 9 resolved · 0 Pith anchors

[1] On the consistency of some partition theorems for continuous colorings, and the structure of ℵ1-dense real order types 1985

[2] All ℵ1-dense sets of reals can be isomorphic 1973

[3] Concerning similarity transformations of linearly ordered sets 1940

[4] Order types and similarity transforma- tions 1955

[5] Entangledness in Suslin lines and trees 2020

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-06-24T14:15:51.847609Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

b1cd40c48e1dd74e85d43119352940a141015375902f1273bce03396b9ef8b9e

Aliases

arxiv: 2507.17503 · arxiv_version: 2507.17503v2 · doi: 10.48550/arxiv.2507.17503 · pith_short_12: WHGUBREODXLU · pith_short_16: WHGUBREODXLU5BOU · pith_short_8: WHGUBREO

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/WHGUBREODXLU5BOUGEMTKKKAUF \
  | 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: b1cd40c48e1dd74e85d43119352940a141015375902f1273bce03396b9ef8b9e

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "8595cd505f199e789e7051ed1d8c174fff4750113cff06624194b9abbd4a5779",
    "cross_cats_sorted": [
      "math.CO"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.LO",
    "submitted_at": "2025-07-23T13:38:59Z",
    "title_canon_sha256": "d48cc478b5e3b0d90ac8cb4354b2292e7a0f727de39864c3e3880881db50c0cc"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2507.17503",
    "kind": "arxiv",
    "version": 2
  }
}