Theoretical analysis verifies the Erdős–Sauer clique decomposition conjecture for 3-graphs up to nine vertices, establishing structural criteria for local tetrahedron packings.
Key Points
To resolve the r=3 case of the Erdős–Sauer clique decomposition conjecture on small hypergraphs and develop a local structural theory for maximum tetrahedron packings.
Applied near-Turán reduction techniques combined with exact Turán numbers to verify 3-graphs through seven vertices.
Analytically evaluated eight-vertex graphs via a Steiner quadruple system S(3,4,8) and nine-vertex graphs via an S(3,4,10) with collision estimates under random relabelling.
Formulated a deterministic local packing framework incorporating leaf-rigidity lemmas, actual-bridge graphs, and pack-cover optimization.
Confirmed that the Erdős–Sauer decomposition bound m - 3ν(G) ≤ ex_3(n, K_4^3) holds analytically for all 3-graphs on up to nine vertices.
Proved that an internal four-pack cover incurs a cost of at most 11, which decreases to 10 under a mild intersection condition.
Formulated a boundary-credit criterion that converts internal cover savings into a valid 3-peeling reduction when residual boundaries have small cover numbers.