Evolutionary families of sets (Q1967111)

This paper was motivated by a conjecture of \textit{D. J. Naddef} and \textit{W. R. Pulleyblank} [Ear decompositions of elementary graphs and \(\text{GF}_2\)-rank of perfect matchings, Ann. Discrete Math. 16, 241-260 (1982; Zbl 0507.05050)] on ``ear decompositions'' of graphs. Call an ordering \((S_1, \ldots, S_n)\) of subsets of \(S\) evolutionary if, for each \(i > 1\), \[ S_i \cap \bigcup_{j=1}^{i-1} S_j \neq \varnothing\quad\text{ and }\quad S_i \cap (S - \bigcup_{j=1}^{i-1} S_j) \neq \varnothing. \] If a family of sets can be so ordered, call it evolutionary. A few preliminary results, and examples and counterexamples, are presented. Then a tree-like notion of a dendritic family of sets is introduced, and all dendritic families are proven to be evolutionary.