Pith Number

pith:3UE2KB32

pith:2026:3UE2KB326I32LKHW4CEDVYWYWP

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A rank function for Fra\"{\i}ss\'{e} classes and the rank property

Carlos L\'opez-Callejas, Jareb Navarro-Castillo

Certain Fraïssé classes realize every countable ordinal as a value of the rank function measuring distance from universality.

arxiv:2604.14461 v3 · 2026-04-15 · math.LO

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Claims

C1strongest claim

We establish RP for three families of classes: those satisfying the free amalgamation property and the full extension property (covering graphs, hypergraphs, and many others); finite tournaments; and finite linear orders. For the latter, we compute the rank of every countable ordinal: if ω^{β₁}·c₁ is the leading Cantor normal form term of α≥ω, then rk(α)=ω·β₁ + ⌊log₂ c₁⌋.

C2weakest assumption

The hereditary classes under consideration satisfy the free amalgamation property together with the full extension property, or consist exactly of finite tournaments or finite linear orders; the rank function is assumed to be well-defined on σF as introduced by Kubiš and Shelah.

C3one line summary

Introduces theory of a rank function measuring distance from Fraïssé universality and proves it realizes every countable ordinal for free-amalgamation classes, tournaments, and linear orders with explicit computation.

Formal links

1 machine-checked theorem link

Receipt and verification

First computed	2026-06-04T01:08:50.138132Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

dd09a5077af237a5a8f6e0883ae2d8b3d56d1b8cbd54080dae32b21735055825

Aliases

arxiv: 2604.14461 · arxiv_version: 2604.14461v3 · doi: 10.48550/arxiv.2604.14461 · pith_short_12: 3UE2KB326I32 · pith_short_16: 3UE2KB326I32LKHW · pith_short_8: 3UE2KB32

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/3UE2KB326I32LKHW4CEDVYWYWP \
  | 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: dd09a5077af237a5a8f6e0883ae2d8b3d56d1b8cbd54080dae32b21735055825

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "f603a6c1546ac500bdc0509e852b8ae0ce2c9208c215e4d63b1fd14b100b6c9a",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
    "primary_cat": "math.LO",
    "submitted_at": "2026-04-15T22:39:21Z",
    "title_canon_sha256": "ba8b465fc4fc44e758c059c9ee56b5514e8428ae1ac841461b014e120bcc410f"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.14461",
    "kind": "arxiv",
    "version": 3
  }
}