Weighted 3-Wise 2-Intersecting Families

Frankl, Péter; Tokushige, Norihide

doi:10.1006/jcta.2002.3282

Cited by 14 publications

(16 citation statements)

References 12 publications

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“…In [7] the following is proved (see the first paragraph of the proof of Proposition 4 on p. 111 in [7]). …”

Section: Proof Of Theoremmentioning

confidence: 83%

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Random walks and multiply intersecting families

Frankl¹,

Tokushige²

2005

Journal of Combinatorial Theory, Series A

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“…In [7] the following is proved (see the first paragraph of the proof of Proposition 4 on p. 111 in [7]). …”

Section: Proof Of Theoremmentioning

confidence: 83%

“…Some basic results concerning the maximum weighted size of multiply intersecting families can be found in [6][7][8]. Among others, the following is proved in [7].…”

Section: Introductionmentioning

confidence: 95%

Random walks and multiply intersecting families

Frankl¹,

Tokushige²

2005

Journal of Combinatorial Theory, Series A

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“…Frankl [8] and Gronau [17][18][19][20] determined s(n, r = 3, t = 1) for n 53. Gronau [18] also proved s(n, r 4, t = 1) = n−1 (n−1)/2 for all n. For sufficiently large n, it was proved that s(n, r 4, t = 2) = n−2 (n−2)/2 in [12], s(n, r, t) = n−t (n−t)/2 for r 5 and 1 t 2 r−2 log 2 − 1 in [29], and s(n, r = 3, t = 2) was determined in [12,14]. Using Theorem 2 we prove the following.…”

Section: Theorem 2 Let (R T) ∈ a Be Fixed Where A Is Defined Bymentioning

confidence: 94%

“…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]. All known results suggest m(n, k, r, t) = max i F i (n, k, r, t) in general, and we will consider the case when the maximum is attained by F 0 or F 1 .…”

Section: Introductionmentioning

confidence: 94%

Multiply-intersecting families revisited

Tokushige

2007

Journal of Combinatorial Theory, Series B

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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 }.

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“…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)|.…”

Section: N Rt) = {G ⊂ [N] : |G ∩ [T + Ri]| ≥ T + (R − 1)i} F I (Nmentioning

confidence: 97%