Delta minors, delta free clutters, and entanglement

For an integer n ≥ 3, the clutter ∆n := {1, 2}, {1, 3}, . . ., {1, n}, {2, 3, . . ., n} is called a delta of dimension n, whose members are the lines of a degenerate projective plane. In his seminal paper on nonideal clutters, Lehman revealed the role of the deltas as a distinct class of minimally nonideal clutters [The width length inequality and degenerate projective planes, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 1, AMS, Providence, RI, 1990, pp. 101-105]. A clutter is delta free if it has no delta minor. Binary clutters, ideal clutters, and clutters with the packing property are examples of delta free clutters. In this paper, we introduce and study basic geometric notions defined on clutters, including entanglement between clutters, a notion that is intimately linked with set covering polyhedra having a convex union. We will then investigate the surprising geometric attributes of delta minors and delta free clutters.