Pith Number

pith:GDVBDKAZ

pith:2026:GDVBDKAZHS4OCKG7CMFZD5R7GU

not attested not anchored not stored refs resolved

Triangles in graphs without the expansion of $4$-cycle

Jialei Song, Long-tu Yuan, Qi Wu

The expansion of the 4-cycle is the only counterexample to a conjecture on the maximum number of triangles in graphs avoiding expanded paths and cycles.

arxiv:2605.17430 v1 · 2026-05-17 · math.CO

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\usepackage{pith}
\pithnumber{GDVBDKAZHS4OCKG7CMFZD5R7GU}

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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 resolve the remaining case C_4^Δ, demonstrating that this is the only counterexample to their conjecture.

C2weakest assumption

The validity of the conjecture for all previously confirmed cases (P_k^Δ for k≥4 and C_k^Δ for k≥5) and the standard definition of the graph expansion F^Δ as replacing each edge by a triangle.

C3one line summary

Resolves the remaining C_4^Δ case of the conjecture on maximum triangles in graphs without P_k^Δ or C_k^Δ for k≥4, showing it is the only counterexample.

References

11 extracted · 11 resolved · 0 Pith anchors

[1] N. Alon and C. Shikhelman. ManyTcopies inH-free graphs.J. Combin. Theory Ser. B, 121:146–172, 2016. 2 2016

[2] P. Erd˝ os. On the number of complete subgraphs contained in certain graphs.Magyar Tud. Akad. Mat. Kutat´ o Int. K¨ ozl., 7:459–464, 1962. 2 1962

[3] D. Gerbner and B. Patk´ os. Generalized Tur´ an results for intersecting cliques.Discrete Math., 347(1):113710, 2024. 2 2024

[4] A. Kostochka, D. Mubayi, and J. Verstra¨ ete. Tur´ an problems and shadows I: Paths and cycles.J. Combin. Theory Ser. A, 129:57–79, 2015. 2, 3 2015

[5] A. Kostochka, D. Mubayi, and J. Verstra¨ ete. Tur´ an problems and shadows II: Trees. J. Combin. Theory Ser. B, 122:457–478, 2017. 2 8 2017

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:03:58.112200Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

30ea11a8193cb8e128df130b91f63f35349cccbf49614a346e2e25c08c4687ee

Aliases

arxiv: 2605.17430 · arxiv_version: 2605.17430v1 · doi: 10.48550/arxiv.2605.17430 · pith_short_12: GDVBDKAZHS4O · pith_short_16: GDVBDKAZHS4OCKG7 · pith_short_8: GDVBDKAZ

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/GDVBDKAZHS4OCKG7CMFZD5R7GU \
  | 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: 30ea11a8193cb8e128df130b91f63f35349cccbf49614a346e2e25c08c4687ee

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "493f6513a411d96f5250d05f8130366fc4a5fd8bb8b317482462e208ae5216ca",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-05-17T12:50:18Z",
    "title_canon_sha256": "f139dcffe9a75d13811f864728c0982e0aa14b1413dee550422e87f4b344a85a"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.17430",
    "kind": "arxiv",
    "version": 1
  }
}