What does this research mean for the field?

The conjectured bound for the Erdős–Sauer clique decomposition problem for 3-graphs holds for all 3-graphs on up to nine vertices, supported by a deterministic local theory around maximum tetrahedron packings. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To verify the Erdős–Sauer clique decomposition conjecture for 3-uniform hypergraphs on up to nine vertices and establish structural local packing reductions.

September 8, 2026Open Access

The Erdős–Sauer Clique Decomposition Problem for 3-Graphs: Verification through Nine Vertices and Local Packing Reductions

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ZRZeraoulia Rafik

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Overview

Theoretical analysis demonstrates the Erdős–Sauer conjecture for 3-graphs with up to nine vertices, suggesting structural pathways toward resolving the general hypergraph decomposition problem.

Key Points

To verify the Erdős–Sauer clique decomposition conjecture for 3-uniform hypergraphs on up to nine vertices and establish structural local packing reductions.
Applied near-Turán reductions with exact Turán numbers for cases with up to seven vertices.
Conducted analytical proofs using Steiner systems S(3,4,8) for eight vertices and S(3,4,10) with random relabelling collision estimates for nine vertices.
Developed a deterministic local theory around maximum tetrahedron packings using actual-bridge graphs, leaf-rigidity lemmas, and boundary-credit criteria.
Proved that the clique decomposition bound m - 3ν(G) ≤ ex_3(n, K_4^3) holds unconditionally for all 3-graphs on at most nine vertices.
Established an internal four-pack cover bound of at most 11 (reduced to 10 under a mild intersection condition) to facilitate 3-peeling steps.

Cite This Study

Zeraoulia Rafik (2026) studied this question.

synapsesocial.com/papers/6aa0091858e84d0ff5b47b0b https://doi.org/10.5281/zenodo.22568303

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September 8, 2026Open Access

The Erdős–Sauer Clique Decomposition Problem for 3-Graphs: Verification through Nine Vertices and Local Packing Reductions

View Full Paper

Ask AI

Bookmark

Authors

ZRZeraoulia Rafik

Discussion

Member takes

Overview

Theoretical analysis demonstrates the Erdős–Sauer conjecture for 3-graphs with up to nine vertices, suggesting structural pathways toward resolving the general hypergraph decomposition problem.

Key Points

To verify the Erdős–Sauer clique decomposition conjecture for 3-uniform hypergraphs on up to nine vertices and establish structural local packing reductions.
Applied near-Turán reductions with exact Turán numbers for cases with up to seven vertices.
Conducted analytical proofs using Steiner systems S(3,4,8) for eight vertices and S(3,4,10) with random relabelling collision estimates for nine vertices.
Developed a deterministic local theory around maximum tetrahedron packings using actual-bridge graphs, leaf-rigidity lemmas, and boundary-credit criteria.
Proved that the clique decomposition bound m - 3ν(G) ≤ ex_3(n, K_4^3) holds unconditionally for all 3-graphs on at most nine vertices.
Established an internal four-pack cover bound of at most 11 (reduced to 10 under a mild intersection condition) to facilitate 3-peeling steps.

Cite This Study

Zeraoulia Rafik (2026) studied this question.

synapsesocial.com/papers/6aa0091858e84d0ff5b47b0b https://doi.org/10.5281/zenodo.22568303

View Full Paper

Ask AI

Bookmark