A 2-(22, 8, 4) design cannot have a 2-(10, 4, 4) subdesign (Q1866018)

A \(t\)-\((v,k,\lambda)\) design is a family of \(k\)-subsets, called blocks, of a \(v\)-set of points, such that each \(t\)-subset of the \(v\)-set is contained in exactly \(\lambda\) blocks. A 2-\((v,k,\lambda)\) design is called a balanced incomplete block design (BIBD). The smallest BIBD whose existence is still undecided is 2-\((22,8,4)\). In this paper, the author shows that a 2-\((22,8,4)\) design cannot have a 2-\((10,4,4)\) subdesign. This result is obtained by classifying all 2-\((10,4,4)\) designs and trying to find 2-\((22,8,4)\) designs by solving instances of the maximum clique problem.