Conjecture on the exact value ex(Q_7, C_4) = 304

Prove that ex(Q_7, C_4) = 304, i.e., establish that the maximum number of edges in a C4-free subgraph of the 7-dimensional hypercube Q_7 equals 304.

Background

The paper constructs and certifies many C4-free subgraphs of Q_7 with 304 edges and finds no 305-edge C4-free subgraph across extensive computational searches, providing empirical evidence for the conjectured value.

The authors enumerate all 672 four-cycles in Q_7 to certify C4-freeness and report 19,866 distinct 304-edge solutions, while 305-edge searches consistently fail to eliminate C4 violations.

References

The consistent failure of $305$-edge searches (minimum $C_4$ violation never reaching $0$ over $1\,076$ independent trials) supports the conjecture $ex(Q_7,C_4)=304$.

— New Lower Bounds for C4-Free Subgraphs of the Hypercubes Q6, Q7, and Q8: Constructions, Structure, and Computational Method  (2603.29127 - Minamoto, 31 Mar 2026) in Abstract; also stated as a Conjecture in Section "Solution Landscape for Q_7"