General determination of ex(Q_n, C_4)

Determine the exact value of ex(Q_n, C_4), the maximum number of edges in a C4-free subgraph of the n-dimensional hypercube Q_n, for general dimension n.

Background

The extremal function ex(Q_n, C_4) equals the maximum number of edges in a subgraph of the n-dimensional hypercube Q_n that contains no 4-cycles. Although exact values are known for small n (e.g., n ≤ 6), the general determination remains unresolved.

The paper improves lower bounds for n = 7 and n = 8, but emphasizes that the overall problem of determining ex(Q_n, C_4) in general is still unsolved, despite classical interest dating back to Erdős and significant progress on asymptotic bounds.

References

The problem of determining $ex(Q_n,C_4)=\max{|E(G)|:G\subseteq Q_n,\,G\text{ is }C_4\text{-free}}$ was raised by Erd\H{o}s and remains open in general.

— 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 Introduction