Fractional L-intersecting Families

Abstract

Let $L = \{\frac{a_1}{b_1}, \ldots , \frac{a_s}{b_s}\}$, where for every $i \in [s]$, $\frac{a_i}{b_i} \in [0,1)$ is an irreducible fraction. Let $\mathcal{F} = \{A_1, \ldots , A_m\}$ be a family of subsets of $[n]$. We say $\mathcal{F}$ is a fractional $L$-intersecting family if for every distinct $i,j \in [m]$, there exists an $\frac{a}{b} \in L$ such that $|A_i \cap A_j| \in \{ \frac{a}{b}|A_i|, \frac{a}{b} |A_j|\}$. In this paper, we introduce and study the notion of fractional $L$-intersecting families.

Maximal fractional cross-intersecting families

“…Frankl and Wilson [9] extended the above result to the non-uniform case by showing that |F | ≤ n 0 + n 1 ...+ n s . More extensions can be found in [2,5,10,11,14,16,17].…”

An ensemble of high rank matrices arising from tournaments

“…Determine min M ∈Mn(a) rank(M ). Here, rank(M ) denotes the rank of M over the field F. This problem first appeared in [2], though it was only stated for matrices over the reals. There it is asked whether one can find an absolute constant c > 0 such that rank(M ) ≥ cn for all M ∈ M n (a).…”

“…The problem of determining the rank of specific matrices has been of immense interest in extremal combinatorics with applications in theoretical computer science as well-see [1,3,4,6,7,9,10,12]. The question in [2] is motivated by a problem in extremal combinatorics concerning what are called self-bisecting families: a family of subsets F of [n] is called a self-bisecting family if, for any distinct A, B ∈ F, either |A∩B| |A| = 1 2 or |A∩B| |B| = 1 2 , and one seeks to find the maximum size of a self-bisecting family of [n]. One of the results that appears in [2] shows that any self-bisecting family has size O(n log 2 n), while there are selfbisecting families of size Ω(n).…”

“…The question in [2] is motivated by a problem in extremal combinatorics concerning what are called self-bisecting families: a family of subsets F of [n] is called a self-bisecting family if, for any distinct A, B ∈ F, either |A∩B| |A| = 1 2 or |A∩B| |B| = 1 2 , and one seeks to find the maximum size of a self-bisecting family of [n]. One of the results that appears in [2] shows that any self-bisecting family has size O(n log 2 n), while there are selfbisecting families of size Ω(n). If the answer to the question in [2] is affirmative, then in fact it is not hard to see that any self-bisecting family has size O(n), and we include that simple argument here for the sake of completeness.…”

“…One of the results that appears in [2] shows that any self-bisecting family has size O(n log 2 n), while there are selfbisecting families of size Ω(n). If the answer to the question in [2] is affirmative, then in fact it is not hard to see that any self-bisecting family has size O(n), and we include that simple argument here for the sake of completeness. Suppose F is a self-bisecting family of [n] of size m and let X m×n be the matrix whose rows are indexed by the members of F and the columns by the elements of [n] defined as follows: for…”