Let \(I\) be a set (of indices). Then, by definition, the disjoint subset relation \((a_i:i\in I)\to(b_i:i\in I)_{ds}\) means that the \(a_i\), \(b_i\) are cardinals with the property that whenever \(A_i\) is a set of cardinality \(a_i\) there exist pairwise disjoint sets \(X_i\in[A_i]^{b_i}\) \((i\in I)\). Families \((X_i:i\in I)\) are called multitransversals of \((A_i: i\in I)\) of size \((b_i:i\in I)\). A transversal [multitransversal] of a family \(F\) of sets is a family of distinct elements [disjoint sets] one from each number of \(F\) [cf. also \S9, pp. 89-97 in reviewer's Thesis, Ensembles ordonnés et ramifiés, Paris (1935) and Publ. Math. Univ. Belgrade 4, 1-138 (1935; Zbl 0014.39401)]. The authors consider 21 statements and prove the mutual equivalence of the statements (1), (2), (3), (4), (5) (Theorem 1), of statements (6), (7), (8), (9), (10) (Theorem 2), of statements (13), (14), (15), (16) (Theorem 3), respectively, and the following main results: '' Theorem 4: Let \(I\) be a set \(a_i\), \(b_i\) be arbitrary cardinals for \(i\in I\). Put \(S=\{a_i:i\in I; b_i\geq i\}\). Then (17) \(\leftrightarrow\) (18) \(\leftrightarrow\) (19) \(\land\) (20) \(\land\) (21), where (17) \((a_i:i\in I)\to(b_i:i\in I)_{ds}\), (18) \((a_i:i\in I)\) has a multitransversal of size \((b_i:i\in I)\); (19) \(\Sigma(i\in I; a_i\leq k)b_i\leq k\) for every cardinal \(k\); (20) \(\omega(S)\land \bar{\lambda}\notin \text{stat}\lambda\) for every weakly inaccessible cardinal \(\lambda\); (21) if \(m< \omega\) and \(m\leq\Sigma(i\in I; a_1=\aleph_0)b_i\), then \(m+\Sigma(i\in I\); \(a_i\leq n)b_i\leq n\) for sufficiently large finite \(n\). ''Notation: For a cardinal \(c\), \(c\) denotes the set of all cardinals \(< c\). For a regular cardinal \(\lambda\), a set \(A\) is stationary on \(\bar{\lambda}\) if \(A\subset\bar{\lambda}\) and for every regressive function \(f\) on \(A\) there exists \(y< \lambda\) such that \(|f^{-1}\{y\}|=\lambda\), \(\text{stat}\lambda\) denotes the system of all sets which are stationary on \(\bar{\lambda}\).