A Note on Kk, k-Cross Free Families

We give a short proof that for any fixed integer k, the maximum number size of a Kk,k-cross free family is linear in the size of the groundset. We also give tight bounds on the maximum size of a Kk-cross free family in the case when F is intersecting or an antichain. Introduction Let F ⊂ 2. Two sets A, B ∈ F cross if 1. A ∩ B 6= ∅. 2. B 6⊂ A and A 6⊂ B. F ⊂ 2 is said to be Kk-cross free if it does not contain k sets A1, ..., Ak such that Ai cross Aj for every i 6= j. Karzanov and Lomonosov conjectured that for any fixed k, the maximum size of a Kk-cross free family F ⊂ 2 [n] is O(n) [5], [1]. The conjecture has been proven for k = 2 and k = 3 [7], [4]. For general k, the best known upperbound is 2(k − 1)n log n, which can easily be seen by a double counting argument on the number of sets of a fixed size. We say that F is Kk,k-cross free if it does not contain 2k sets A1, ..., Ak, B1, ..., Bk ∈ F such that Ai crosses Bj for all i, j. In this paper, we prove the following: Theorem 1: Let F ⊂ 2 be a Kk,k-cross free family. Then |F| ≤ (2k − 1) n. In this section, we give upperbounds on the maximum size of certain classes of Kkcross free families. By applying Dilworth’s Theorem [2], one can obtain a tight bound the electronic journal of combinatorics 15 (2008), #N39 1 for intersecting k-cross free families. Recall a family F ⊂ 2 is intersecting if for every A, B ∈ F , A ∩ B 6= ∅. Theorem 2: Let F ⊂ 2 be a family that is k-cross free and intersecting. Then |F| ≤ (k − 1)n, and this bound is asymptotically tight. We also obtain tight bounds for Kk-cross free families that is an antichain. Recall F is an antichain if no set in F is a subset of another. Theorem 3: For k ≥ 3, let F ⊂ 2 be a family that is k-cross free and an antichain. Then |F| ≤ (k − 1)n/2, and this bound is asymptotically tight. We define sub(A) to be the number of subsets of A in F . Our next Theorem gives a non-trivial upperbound on a Kk-cross free family based on the number of subsets in each set of our family. Theorem 4: Let F ⊂ 2 be a Kk-cross free family and let m be defined as m = 