Necessary existence criteria for mandatory representation designs \(\text{MRD}(4,k;v)\) (Q2748462)

A mandatory representation design \(\text{MRD}(K;v)\) is a pair \((X,\text{QTR}{\mathcal B})\), where \(X\) is a set of \(v\) points and \(\text{QTR}{\mathcal B}\) is a collection of subsets of \(X\) called blocks, such that each pair of distinct points from \(X\) occurs in a unique block, each block has size an integer in the set \(K,\) and for each \(k\in K\) there is at least one block in \(\text{QTR}{\mathcal B}\) of size \(k.\) MRDs have been extensively studied in the case \(K=\{3,k\}.\) In this paper the authors investigate necessary conditions for the existence of MRDs with \(K=\{4,k\}.\)