Regular (2, 2)-systems

Let (E,I) be an independence system over the finite setE = {e₁, ,e_n}, whose elements are orderede₁ e_n. (E,I) is called regular, if the independence of {e_l, ,e_lk},l₁ < <l_k, implies that of {e_ml, ,e_mk}, wherem_l < ··· <m_k andl₁ m₁, ,l_k m_k. (E,I) is called a 2-system, if for anyI I,e E I the setI {e } contains at most 2 distinct circuitsC, C I and the number 2 is minimal with respect to this property. If, in addition, for any two independent setsI andJ the family (C J, C C(J, I)), whereC(J, I) denotes {C C:e J I C {e}}, can be partitioned into 2 subfamilies each of which possesses a transversal, then (E,I) is called a (2, 2)-system. In this paper we characterize regular 2-systems and we show that the classes of regular 2-systems resp. regular (2, 2)-systems are identical.