On Sperner families in which no K sets have an empty intersection

Sperner type theorems and complexity of minimal disjunctive normal forms of monotone boolean functions

“…An i m p r o v e m e n t of our procedure in proving Theorem 4 as the a u t h o r has done in [7] w o u l d improve our n o , but there a r e , for at least # = 3 and small n, functions whose minimal disjunctive normal forms have greater complexity (for certain measures) as t h a t according to Theorem 4. L e t us consider the following ~,XAMPL~: L e t # = 3, let n = 10 and let e = 1 in our measure.…”

“…Using (3) we have according to (6). Repeated applications of this operation yield eventually a Spernsr family ~ in which no # sets have the union R, consisting only of sets having at most a = [(n --1)]2] elements, and its complexity is not less than that of ~.…”

Weighted 3-Wise 2-Intersecting Families

“…For 3-wise 1-intersecting families, it was the subject of several papers of Frankl [3] and Gronau [10,11,12,13] and it is known that for n553 the only optimal families are [ f½n À 1g [ f½n À fn À 1gg;…”

Multiply-intersecting families revisited

“…The case r = 2 was observed by Erdős-Ko-Rado [7], Frankl [9], Wilson [34], and then m(n, k, 2, t) = max i |F i (n, k, 2, t)| was finally proved by Ahlswede and Khachatrian [2]. Frankl [8] showed m(n, k, r, 1) = |F 0 (n, k, r, 1)| if (r − 1)n rk, see also [20,27]. Partial results for the cases r 3 and t 2 are found in [12,14,[29][30][31][32].…”