Mubayi's conjecture states that if\scrF is a family of k-sized subsets of [n] = {1, . . ., n} which, for k ≥ d ≥ 3 and n ≥ dk/d-1, satisfies A1 ∩ ... ∩ Ad ≠ Ø whenever |A1 ∪ .... ∪ Ad| ≤ 2k for all distinct sets A1, ..., Ad ∊ F, then |F| ≤ (k-1), with equality occurring only if F is the family of all k-sized subsets containing some fixed element. This paper proves that Mubayi's conjecture is true for all families that are invariant with respect to shifting; indeed, these families satisfy a stronger version of Mubayi's conjecture. Relevant to the conjecture, we prove a fundamental bijective duality between what we call (i, j)-unstable families and (j, i)-unstable families. Generalizing previous intersecting conditions, we introduce the (d, s, t)-conditionally intersecting condition for families of sets and prove general results thereon. We prove fundamental theorems on two (d, s)-conditionally intersecting families that generalize previous intersecting families, and we pose an extension of a previous conjecture by Frankl and Füredi. Finally, we generalize a classical result by ErdÕs, Ko, and Rado by proving tight upper bounds on the size of (2, s)-conditionally intersecting families F ⊆ 2n-1), with equality occurring only if F is the family of all k-sized subsets containing some fixed element. This paper proves that Mubayi's conjecture is true for all families that are invariant with respect to shifting; indeed, these families satisfy a stronger version of Mubayi's conjecture. Relevant to the conjecture, we prove a fundamental bijective duality between what we call (i, j)-unstable families and (j, i)-unstable families. Generalizing previous intersecting conditions, we introduce the (d, s, t)-conditionally intersecting condition for families of sets and prove general results thereon. We prove fundamental theorems on two (d, s)-conditionally intersecting families that generalize previous intersecting families, and we pose an extension of a previous conjecture by Frankl and Füredi. Finally, we generalize a classical result by ErdÕs, Ko, and Rado by proving tight upper bounds on the size of (2, s)-conditionally intersecting families F ⊆ 2[n] and by characterizing the families that attain these bounds. We extend this theorem for sufficiently large n to families F ⊆ 2[n] whose members have at most a fixed size u. [ABSTRACT FROM AUTHOR]