Multiply-intersecting families revisited

Abstract

Motivated by the Frankl's results in [P. Frankl, Multiply-intersecting families, J. Combin. Theory B 53 (1991) 195-234], we consider some problems concerning the maximum size of multiply-intersecting families with additional conditions. Among other results, we show the following version of the Erdős-KoRado theorem: for all r 5 and 1 t 2 r+1 − 3r − 1 there exist positive constants ε and n 0 such that if n > n 0 and | k n − 1 2 | < ε then r-wise t-intersecting k-uniform families on n vertices have size at most max{ n−t k−t , (t + r) n−t−r k−t−r+1 + n−t−r k−t−r }.

Multiply union families in Nn

“…Much research has been done for the case of families in {0, 1} n , and there are many challenging open problems. The interested reader is referred to [2,3,4,8,9].…”

A product version of the Erdős–Ko–Rado theorem

“…The problem of determining m(n, k, r, t) goes back to Erdős, Ko and Rado [4], and is still wide open. All known results, e.g., [1,[4][5][6][7]12,[22][23][24][25][26][27]29], suggest m(n, k, r, t) = max A (n, k, r, t) .…”

“…But Theorem 8 would be useful to prove (2) (if true) or its product version for the cases 1 as well. In fact, this technique was used in [25][26][27] to get partial results of (2).…”

The Random Walk Method for Intersecting Families

“…(But G 1 is not always optimal for w 0 , for example, we have w 0 (n, p, r, 1) > w p (G 1 (n, r, 1)) if p > 1 2 and r ≤ 5, see [28].) More results for m 0 (n, k, r,t) with k/n ≈ 1/2, and w 0 (n, p, r,t) with p ≈ 1/2 are found in [17,28,29].…”

“…Frankl [8] showed m(n, k, r, 1) = |F 0 (n, k, r, 1)| if (r − 1)n ≥ rk. Partial results for the cases r ≥ 3 and t ≥ 2 are found in [14,16,24,26,27,23,29]. All known results suggest m(n, k, r,t) = max i |F i (n, k, r,t)|.…”