On Mubayis Conjecture and conditionally intersecting sets

Mubayis Conjecture states that if $mathcal{F}$ is a family of $k$-sized subsets of $[n] = {1,ldots,n}$ which, for $k geq d geq 2$, satisfies $A_1 capcdotscap A_d eq emptyset$ whenever $|A_1 cupcdotscup A_d| leq 2k$ for all distinct sets $A_1,ldots,A_d inmathcal{F}$, then $|mathcal{F}|leq binom{n-1}{k-1}$, with equality occurring only if $mathcal{F}$ is the family of all $k$-sized subsets containing some fixed element. This paper proves that Mubayis Conjecture is true for all families that are invariant with respect to shifting; indeed, these families satisfy a stronger version of Mubayis Conjecture. Relevant to the conjecture, we prove a fundamental bijective duality between $(i,j)$-unstable families and $(j,i)$-unstable families. Generalising previous intersecting conditions, we introduce the $(d,s,t)$-conditionally intersecting condition for families of sets and prove general results thereon. We conjecture on the size and extremal structures of families $mathcal{F}inbinom{[n]}{k}$ that are $(d,2k)$-conditionally intersecting but which are not intersecting, and prove results related to this conjecture. We prove fundamental theorems on two $(d,s)$-conditionally intersecting families that generalise previous intersecting families, and we pose an extension of a previous conjecture by Frankl and Furedi on $(3,2k-1)$-conditionally intersecting families. Finally, we generalise a classical result by ErdH{o}s, Ko and Rado by proving tight upper bounds on the size of $(2,s)$-conditionally intersecting families $mathcal{F}subseteq 2^{[n]}$ and by characterising the families that attain these bounds. We extend this theorem for certain parametres as well as for sufficiently large families with respect to $(2,s)$-conditionally intersecting families $mathcal{F}subseteq 2^{[n]}$ whose members have at most a fixed number $u$ members.