Robust but Not Strict: Two Notions of EKR Robustness in Finite Permutation Groups

We study two distinct robustness notions for Erdős–Ko–Rado (EKR) theorems on finite permutation groups, both defined by deleting inverse-closed sets of derangements from the associated Cayley graph. The main strict-robustness results classify the natural projective-line action of PGL(2, q) (strictly robust exactly for q = 2) and the natural action of AGL(d, 2) (strictly robust exactly for d = 1). We also give exact formulae for the ordinary label threshold in the natural actions of Sym(n) and Alt(n), and prove a lower bound for families within a quantified distance of a canonical star in any 2-transitive action. Exact finite checks settle several small degrees, while the ordinary-robustness questions for sufficiently large symmetric and alternating groups remain open. For a separate minimum-label problem for PGL(2, q), we record exact values at q = 3, 4, 5 and unresolved intervals at q = 7, 8, 9. Finite computation, solver output, and proof are identified separately throughout.