For positive integers r and n with r ≤ n, let Pn,r be the family of all sets {(x 1 , y 1 ), . . . , (xr, yr)} such that x 1 , . . . , xr are distinct elements of [n] := {1, . . . , n} and y 1 , . . . , yr are also distinct elements of [n]. Pn,n describes permutations of [n]. For r < n, Pn,r describes r-partial permutations of [n]. Families A 1 , . . . , A k of sets are said to be cross-intersecting if, for any distinct i and j in [k], any set in A i intersects any set in A j . A sharp bound for the sum of sizes of cross-intersecting subfamilies of Pn,n has recently been established by the author. We generalize this bound by showing that, if A 1 , . . . , A k are cross-intersecting subfamilies of Pn,r, thenWe also determine the structures for which the bound is attained when r < n. Our main tool is an extension of Katona's cyclic permutation method.

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We say that a set A t-intersects a set B if A and B have at least t common elements. A family A of sets is said to be t-intersecting if each set in A t-intersects any other set in A. Families A 1 , A 2 , ..., A k are said to be cross-t-intersecting if for any i and j in {1, 2, ..., k} with i = j, any set in A i t-intersects any set in A j . We prove that for any finite family F that has at least one set of size at least t, there exists an integer κ ≤ |F| such that for any k ≥ κ, both the sum and the product of sizes of any k cross-t-intersecting sub-families A 1 , ..., A k (not necessarily distinct or non-empty) of F are maxima if A 1 = ... = A k = L for some largest t-intersecting sub-family L of F. We then study the smallest possible value of κ and investigate the case k < κ; this includes a cross-intersection result for straight lines that demonstrates that it is possible to have F and κ such that for any k < κ, the configuration A 1 = ... = A k = L is neither optimal for the sum nor optimal for the product. We also outline solutions for various important families F, and we provide solutions for the case when F is a power set.

BackgroundLittle is known about differences between mini‐implants and conventional immediately loaded implants for overdentures.ObjectivesTo compare clinical outcomes using two immediately loaded conventional or mini‐implants for mandibular overdentures.Materials and MethodsEdentulous patients receiving either conventional (4.1 mm) or mini‐implants (2.9 mm or less), based on available bone width were analyzed. All implants were immediately loaded with mandibular overdentures installed using locator attachments. Digital periapical radiographs for measuring marginal bone loss and clinical outcomes (ie, periodontal probing, plaque, and bleeding indices) were assessed at 1, 3, 6, and 12‐month follow‐up periods.ResultsFifty patients (25 receiving conventional implants—12 females, mean age of 65.3 ± 7.3 years; and 25 receiving mini‐implants—11 females, mean age of 66.8 ± 8.1 years) was analyzed. Peak insertion torque (P = .001) and bone loss (P = .02), as well as change in plaque (P = .02) and bleeding (P = .04) indices at 12 months differed significantly between groups. Furthermore, linear regression revealed the height of the locator as a risk factor for bone loss (P = .038).ConclusionsThe present findings suggest that two mini‐implants are significantly more susceptible to bone loss after immediate loading, for which the height of locator might be considered a risk factor.