Pith Number

pith:UPY6OK5K

pith:2026:UPY6OK5KCQM7DXKMAXPKM3ZEUL

not attested not anchored not stored refs pending

Saturation numbers for joins of graphs and characterization of extremal graphs

Xinying Hua, Yuejian Peng

arxiv:2606.22011 v1 · 2026-06-20 · 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{UPY6OK5KCQM7DXKMAXPKM3ZEUL}

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.

Receipt and verification

First computed	2026-06-23T02:13:05.513480Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

a3f1e72baa1419f1dd4c05dea66f24a2c7e6e23895cb4cc7044919d17501d2ee

Aliases

arxiv: 2606.22011 · arxiv_version: 2606.22011v1 · doi: 10.48550/arxiv.2606.22011 · pith_short_12: UPY6OK5KCQM7 · pith_short_16: UPY6OK5KCQM7DXKM · pith_short_8: UPY6OK5K

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/UPY6OK5KCQM7DXKMAXPKM3ZEUL \
  | 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: a3f1e72baa1419f1dd4c05dea66f24a2c7e6e23895cb4cc7044919d17501d2ee

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "07632d2d998c6f81d9675241d0548c62bf6b4456808e76548c13dda2a7344cd1",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-06-20T12:20:03Z",
    "title_canon_sha256": "725512b5c539d351fe3963390dd403b4ba3cbc106ef51aed7b7fa8eab4ddf09e"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2606.22011",
    "kind": "arxiv",
    "version": 1
  }
}