Pith Number

pith:X7YPFKRA

pith:2026:X7YPFKRACFEF663PWZT7TEHC7R

not attested not anchored not stored refs resolved

Cover-free families on graphs

Lucia Moura, Prangya Parida

For any simple graph G, the minimum t for a G-Sperner family equals t(1, χ(G)).

arxiv:2605.12634 v1 · 2026-05-12 · math.CO

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\pithnumber{X7YPFKRACFEF663PWZT7TEHC7R}

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2 Internet Archive

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

We prove t_s(G) = t(1, χ(G)) for any simple graph G.

C2weakest assumption

The graph is simple with no isolated vertices when applying the trivial bound t(1,n) ≤ t(G) ≤ t(2,n); constructions for specific graphs assume standard combinatorial encodings like mixed-radix representations exist without hidden overlaps.

C3one line summary

For any graph G, the minimum universe size for G-Sperner families equals the chromatic number of G, while G-cover-free families on paths and cycles satisfy log2(n) ≤ t ≤ 1.893 log2(n) + O(1).

References

29 extracted · 29 resolved · 0 Pith anchors

[1] Linear time constructions of some-restriction problems 2015

[2] An efficient algorithm for group testing with runlength constraints.Discrete Applied Mathematics, 360:181–187, 2025 2025

[3] A decomposition theorem for partially ordered sets.Classic papers in combinatorics, pages 139–144, 1987 1987

[4] Dingzhu Du and Frank Hwang.Combinatorial group testing and its applications, volume 12. World Scientific, 2000 2000

[5] Bounds on the length of disjunctive codes.Problemy Peredachi Informatsii, 18(3):7–13, 1982 1982

Receipt and verification

First computed	2026-05-18T03:10:00.116258Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

bff0f2aa2011485f7b6fb667f990e2fc69ae67fe1fc50987757bcd7e3bd52609

Aliases

arxiv: 2605.12634 · arxiv_version: 2605.12634v1 · doi: 10.48550/arxiv.2605.12634 · pith_short_12: X7YPFKRACFEF · pith_short_16: X7YPFKRACFEF663P · pith_short_8: X7YPFKRA

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Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/X7YPFKRACFEF663PWZT7TEHC7R \
  | 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: bff0f2aa2011485f7b6fb667f990e2fc69ae67fe1fc50987757bcd7e3bd52609

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "0766cacf3d6fd28ad3e4c41e0611eeb3aebaa9df8ca67e2a05c5e821fbacacd6",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-05-12T18:17:08Z",
    "title_canon_sha256": "0d9e877305faaa646502f2c9167c57e271dedf23d2d49533f7a3d11ed8b8c188"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.12634",
    "kind": "arxiv",
    "version": 1
  }
}