A Note on Indecomposable Kirkman Squares

A \(KS_2(v;1,\lambda)\) is called indecomposable if it is not isomorphic to the direct sum of a \(KS_2(v;1,\lambda_1)\) with a \(KS_2(v ;1,\lambda_2)\) for some \(\lambda_1\) and \(\lambda_2\) which add to \(\lambda\). In this note, we show that there exists an indecomposable \(KS_2(v;1,\lambda)\) for \(v \equiv 0 \pmod{2}\), \(v \geq 4\), and \(\lambda \geq 2\).