Weak Saturation of the Cube in the Clique: Exact Values for n ≤ 13 and Linear Bounds

For graphs \(G\) and \(H\), \(\operatorname{wsat}(G,H)\) is the least number of edges of a spanning subgraph \(F\) of \(G\) such that the edges of \(G\) missing from \(F\) can be added one at a time, each completing a new copy of \(H\). Morrison and Noel asked (Open Problem Garden, 2016) to determine \(\operatorname{wsat}(K_n,Q_3)\), where \(Q_3\) is the cube. We give a partial answer. For \(n\le7\) no edge can be added and the value is \(\binom n2\). By exhaustive computer searches, each carried out with at least two independent programs, \(\operatorname{wsat}(K_n,Q_3)=15,16,18,19,21,23\) for \(n=8,9,10,11,12,13\); extremal graphs are given explicitly, with certificates that can be checked without a computer. The values for \(n\le11\) and the upper bound \(\operatorname{wsat}(K_n,Q_3)\le\lfloor (7n+2)/4\rfloor\) for \(n\ge9\), which comes from a gadget with four vertices and seven edges, were obtained earlier in two unrefereed preprints (Carptopus, 2026; V. Saha and N. Saha, 2026); the values for \(n=12,13\) are new. We prove \(\operatorname{wsat}(K_n,Q_3)\ge(5n-7)/3\) for \(n\ge8\) by the cluster method of Terekhov and Zhukovskii, with clusters that may share a vertex. Hence the limit of \(\operatorname{wsat}(K_n,Q_3)/n\), which exists by a result of Alon, lies in \([5/3,7/4]\); the lower end improves the value \(11/7\) given by a general theorem of Terekhov and Zhukovskii. We conjecture, on computational evidence only, that \(\operatorname{wsat}(K_n,Q_3)=\lfloor (7n+2)/4\rfloor\) for all \(n\ge9\). The exact value for \(n\ge14\) remains open. This is an unrefereed note.