Fractional L-intersecting families

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 \emph{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.


Introduction
Let [n] denote {1, . . . , n} and let L = {l 1 , . . . , l s } be a set of s non-negative integers. A family F = {A 1 , . . . , A m } of subsets of [n] is L-intersecting if for every A i , A j ∈ F , A i = A j , |A i ∩A j | ∈ L. In 1975, it was shown by Ray-Chaudhuri and Wilson in [13] that if F is t-uniform, then |F | ≤ n s . Setting L = {0, . . . , s − 1}, the family F = [n] s is a tight example to the above bound, where [n] s denotes the set of all s-sized subsets of [n]. In the non-uniform case, it was shown by Frankl and Wilson in the year 1981 (see [7]) that if we don't put any restrictions on the cardinalities of the sets in F , then |F | ≤ n s + n s−1 + · · · + n 0 . This bound is tight as demonstrated by the set of all subsets of [n] of size at most s with L = {0, . . . s − 1}. The proof of this bound was using the method of higher incidence matrices. Later, in 1991, Alon, Babai, and Suzuki in [2] gave an elegant linear algebraic proof to this bound. They showed that if the cardinalities of the sets in F belong to the set of integers K = {k 1 , . . . , k r } with every k i > s − r, then |F | is at most n s + n s−1 + · · · + n s−r+1 . The collection of all the subsets of [n] of size at least s − r + 1 and at most s with K = {s − r + 1, . . . , s} and L = {0, . . . , s − 1} forms a tight example to this bound. In 2002, this result was extended by Grolmusz and Sudakov [8] to k-wise L-intersecting families. In 2003, Snevily showed in [14] that if L is a collection of s positive integers then |F | ≤ n−1 s + n−1 s−1 + · · · + n−1 0 . See [11] for a survey on L-intersecting families and their variants.
In this paper, we introduce a new variant of L-intersecting families called the fractional Lintersecting families. Let L = { a 1 b 1 , . . . , as bs }, where for every i ∈ [s], a i b i ∈ [0, 1) is an irreducible fraction. Let F = {A 1 , . . . , A m } be a family of subsets of [n]. We say F is a fractional Lintersecting family if for every distinct i, j ∈ [m], there exists an a b ∈ L such that and therefore (using the result in [13]), |F | ≤ n s . A tight example to this bound is given by the So what is interesting is finding a good upper bound for |F | in the non-uniform case. Unlike in the case of the classical L-intersecting families, it is clear from the above definition that if A and B are two sets in a fractional L-intersecting family, then the cardinality of their intersection is a function of |A| or |B| (or both).
In Section 2.1, we prove the following theorem which gives an upper bound for the cardinality of a fractional L-intersecting family in the general case. We follow the convention that a b is 0, when b > a.
Consider the following examples for a fractional L-intersecting family. 3 4 , . . . , 1 n , . . . , n−1 n }, where we omit fractions, like 2 4 , which are not irreducible. The collection of all the non-empty subsets of [n] is a fractional L-intersecting family of cardinality 2 n − 1. Here, |L| = s ∈ Θ(n 2 ). Since t ≥ s, we can apply Statement (b) of Theorem 1 to get an upper bound of c 1 (2 n − 1) which is asymptotically tight. In general, when where c ≥ 0 is a constant, the set of all the non-empty subsets of [n] of cardinality at most n − c is an example which demonstrates that the bound given in Statement (b) of Theorem 1 is asymptotically tight. is a fractional L-intersecting family of cardinality n s . In this case, the bound given by Theorem 1 is asymptotically tight up to a factor of ln 2 n ln ln n . We believe that if F is a fractional L-intersecting family of maximum cardinality, where s (= |L|) is a constant, then |F | ∈ Θ(n s ).
Coming back to the classical L-intersecting families, it is known that when F is an Lintersecting family where |L| = s = 1, the Fisher's Inequality (see Theorem 7.5 in [9]) yields |F | ≤ n. Study of such intersecting families was initiated by Ronald Fisher in 1940 (see [5]). This fundamental result of design theory is among the first results in the field of L-intersecting families. Analogously, consider the scenario when L = { a b } is a singleton set. Can we get a tighter (compared to Theorem 1) bound in this case? We show in Theorem 2 that if b is a constant prime we do have a tighter bound. 2 −2 thereby implying that the bound obtained in Theorem 2 is asymptotically tight up to a factor of ln n when b is a constant prime. We believe that the cardinality of such families is at most cn, where c > 0 is a constant.
The rest of the paper is organized in the following way: In Section 2.1, we give the proof of Theorem 1 after introducing some necessary lemmas in the beginning. In Theorem 6 in Section 2.2, we give an upper bound of n for fractional L-intersecting families on [n] whose member sets are 'large enough'. In Section 3, we consider the case when L is a singleton set and give the proof of Theorem 2. Later in this section, in Theorem 8, we consider the case when the cardinalities of the sets in the fractional L-intersecting family are restricted. Finally, we conclude with some remarks, some open questions, and a conjecture.

The general case 2.1 Proof of Theorem 1
Before we move to the proof of Theorem 1, we introduce a few lemmas that will be used in the proof.

Few auxiliary lemmas
The following lemma is popularly known as the 'Independence Criterion' or 'Triangular Criterion'.
Then f 1 , . . . , f m are linearly independent members of the space F Ω .
Proof. Arrange every subset of [n] of cardinality less than p in a linear order, say ≺, such that A ≺ B implies |A| ≤ |B|. For any two distinct sets A and B, we know that 0 has a non-trivial solution. Then, identify the first set, say A 0 , in the linear order ≺ for which λ A 0 is non-zero. Evaluate the functions on either side of the above equation at V A 0 to get λ A 0 = 0 which is a contradiction to our assumption.

The proof
Proof. Let p be a prime and let p > t. We partition F into p parts, namely F 0 , . . . , Estimating |F i |, when i > 0.
, in the following way.
for any positive integer r. Each f j is thus an appropriate linear combination of distinct monomials of degree at most s. Therefore, We can improve this bound by using the "swallowing trick" in a way similar to the way it is used in the proof of Theorem 1.1 in [2].
In order to prove the claim, assume m j=1 Evaluating at V j , all terms in the second sum vanish (since f (V j ) = 0) and by Equation 1, only the term with subscript j remains of the first sum. We infer that λ j = 0, for every j. It then follows from Lemma 4 that every µ A is zero thus proving the claim.
Since each function in the collection of functions in Claim 5.1 can be obtained as a linear combination of distinct monomials of degree at most s, we can infer that m + s−1 j =i,j=0 n j ≤ s j=0 n j . We thus have Observe that i ≤ p − 1. We will shortly see that the prime p we choose is always at most 2g(t, n) ln(g(t, n)), where g(t, n) = (2t+ln n) ln(2t+ln n) . So if s ≤ n + 1 − 2g(t, n) ln(g(t, n)), the condition s+i ≤ n (here i stands for the symbol k in Lemma 5) given in Lemma 5 is satisfied and therefore the more powerful Lemma 5 can be used instead of Lemma 4 while applying the swallowing trick. We can then claim that (proof of this claim is similar to the proof of Claim 5.1 and is therefore omitted is a collection of functions that is linearly independent in the vector space F {0,1} n p over F p which can be obtained as a linear combination of distinct monomials of degree at most s. It then follows that |F i | ≤ n s . In the rest of the proof, we shall assume the general bound for |F i | given by Inequality 2. (Using the n s upper bound for |F i | in place of Inequality 2 when s ≤ n + 1 − 2g(t, n) ln(g(t, n)) in the rest of the proof will yield the tighter bound for |F | given in Statement (a) in the theorem.) Observe that we still do not have an estimate of |A 0 | since i ≡ a l b l i (mod p) when i ≡ 0 (mod p). To overcome this problem, consider the collection P = {p q+1 , . . . , p r } of r − q smallest primes with p q ≤ t < p q+1 < · · · < p r (p j denotes the j-th prime; p 1 = 2, p 2 = 3, and so on) such that for every A ∈ F , there exists a prime p ∈ P with p ∤ |A|. Note that if we repeat the steps done above for each p ∈ P , we obtain the following upper bound.
To obtain a small cardinality set P of the desired requirement, we choose the minimum r such that p q+1 p q+2 · · · p r > n. If t > n − c 1 , for some positive integer constant c 1 , then P = {p q+1 , . . . , p q+c 1 } satisfies the desired requirements of P . We thus have, The product of the first k primes is the primorial function p k # and it is known that p k # = e (1+o(1))k ln k . Given a natural number N, let N# denote the product of all the primes less than or equal to N (some call this the primorial function). It is known that N# = e (1+o(1))N . Since pr# t# = p k+1 p k+2 · · · p r , setting e (1+o(1))r ln r e (1+o(1))t > n, we get, r ≤ 2(2t+ln n) ln(2t+ln n) = g(t, n). Using the prime number theorem, the rth prime p r is at most 2r ln r. Thus, we have p r ≤ 2g(t, n) ln(g(t, n)). Substituting for r and p r in Inequality 3 gives the theorem.

When the sets in F are 'large enough'
In the following theorem, we show that when the sets in a fractional L-intersecting F are 'large enough', then |F | is at most n. For every A i ∈ F , we define its (+1, −1)-incidence vector as: We prove the theorem by proving the following claim.
Claim 6.1. X A 1 , . . . , X Am are linearly independent in the vector space R n over R.
Assume for contradiction that X A 1 , . . . , X Am are linearly dependent in the vector space R n over R. Then, we have some reals λ A 1 , . . . , λ Am where not all of them are zeroes such that It is given that, for every A i ∈ F , |A i | > n 2 . Let u = (1, 1, . . . , 1) ∈ R n be the all ones vector. Then, X A i , u > 0, for every A i ∈ F . Therefore, if all non-zero λ A i s in Equation (5) are of the same sign, say positive, then the inner product of u with the L.H.S of Equation (5) would be non-zero which is a contradiction. Hence, we can assume that not all λ A i s are of the same sign. We rewrite Equation (5) by moving all negative λ A i s to the R.H.S. Without loss of generality, assume λ A 1 , . . . , λ A k are non-negative and the rest are negative. Thus, we have where v is a non-zero vector.
For any two distinct sets A, B ∈ F , ∃ a i b i ∈ L such that Since Applying the fact that the cardinality of every set S in F satisfies αn < |S| ≤ n, which is a contradiction. This proves the claim and thereby the theorem.

L is a singleton set
As explained in Section 1, the Fisher's Inequality is a special case of the classical L-intersecting families, where |L| = 1. In this section, we study fractional L-intersecting families with |L| = 1; a fractional variant of the Fisher's inequality. From the definition of a fractional a b -intersecting family it is clear that |H| ≤ 1. The rest of the proof is to show that |F | ≤ (b − 1)(n + 1)⌈ ln n ln b ⌉. We do this by partitioning F into (b − 1)⌈log b n⌉ parts and then showing that each part is of size at most n + 1. We define F j i as

Proof of Theorem 2
, create a vector X A as follows: Let |F ib k−1 b k | = m. Let M k,i denote the m × n matrix formed by taking X A s as rows for each A ∈ F ib k−1 b k . Then, |F ib k−1 b k | ≤ n + 1 can be proved by considering B = M k,i × M T k,i and showing that B − aiJ (, where J is the m × m all 1 matrix, ) has full rank; determinant of B − aiJ is non-zero since the only term not divisible by the prime b in the expansion of its determinant comes from the product of all the diagonals (note that a < b, i < b, and since b is a prime, we have b ∤ ai).
We shall call F a bisection closed family if F is a fractional L-intersecting family where L = { 1 2 }. We have two different constructions of families that are bisection closed and are of cardinality 3n 2 − 2 on [n].
where H(0) = 1. Now consider the matrix: Let M ′ (k) be the matrix obtained from M(k) by removing the first and the (2 k + 1)th rows and replacing the -1's by 1's and 1's by 0's. M ′ (k) is clearly bisection closed and has cardinality 3n 2 − 2, where n = 2 k .

Restricting the cardinalities of the sets in F
When L = { a b }, where b is a prime, Theorem 2 yields an upper bound of O( b log b n log n) for |F |. However, we believe that when |L| = 1, the cardinality of any fractional L-intersecting family on [n] would be at most cn, where c > 0 is a constant. To this end, we show in Theorem 8 that when the sizes of the sets in F are restricted, we can achieve this.
The following lemma is crucial to the proof of Theorem 8.  Proof. For any A ∈ F , let Y A ∈ R n be a vector defined as: Clearly, Y A , Y A = 1. For any two distinct sets A, B ∈ F , we have

Discussion
In Theorem 1, we gave a general upper bound for |F |, where F is a fractional L-intersecting family. In Section 1, we also gave an example to show that this bound is asymptotically tight up to a factor of ln 2 n ln ln n , when s (= |L|) is a constant. However, when s is a constant, we believe that |F | ∈ Θ(n s ).
Consider the following special case for a fractional L-intersecting family F , where L = { 1 2 }. We call such a family a bisection-closed family (see definition in Section 3).

Conjecture 9.
If F is a bisection-closed family, then |F | ≤ cn, where c > 0 is a constant.
We have not been able to find an example of a bisection-closed family of size 2n or more. The problem of determining a linear sized upper bound for the size of any bisection-closed family leads us to pose the following question: Open problem 10. Suppose 0 < a 1 ≤ · · · ≤ a n are n distinct reals. Let M n (a 1 , . . . , a n ) denote the set of all symmetric matrices M satisfying m ij ∈ {a i , a j } for i = j and m ii = 0 for all i. Then, does there exist an absolute constant c > 0 such that rk(M) ≥ cn, for all M ∈ M n (a 1 , . . . , a n )?
To see how this question ties in with our problem, suppose that a family F ⊂ P([n]) is a bisection closed family, i.e., for A, B ∈ F and A = B then |A ∩ B| ∈ {|A|/2, |B|/2}. For simplicity, let us write F = {A 1 , . . . , A m } and denote |A i | = a i where the a i are arranged in ascending order. We say A bisects B if |A ∩ B| = |B|/2. For each A ∈ F , let u A ∈ R n where u A (i) = 1 if i ∈ A and −1 if i ∈ A. Then note that Consider the m × m matrix M whose rows and columns are indexed by the members of F , with M A,B = u A , u B . Then, since M is a Gram matrix of vectors in R n , it follows that rk(M) ≤ n.
If X = 1 2 (nJ − M), where J is the all ones matrix of order m, then rk(X ) ≤ n + 1. But note that X ∈ M(a 1 , . . . , a m ). So, if the answer to the aforementioned open problem is 'yes', then rk(X ) ≥ cm. This gives cm ≤ r(X ) ≤ n + 1 which in turn gives m ≤ c −1 (n + 1).
The problem of determining the maximum size of a fractional L-intersecting family is far from robust in the following sense. Suppose L = {1/2} and we consider the problem of determining the size of an 'ε-approximately fractional L-intersecting family,' i.e., for any A = B we have that at least one of |A∩B| |A| , |A∩B| |B| ∈ (1/2 − ε, 1/2 + ε) for small ε > 0, then such families can in fact be exponentially large in size. Let each set A i be chosen uniformly and independently at random from P([n]). Then since each |A i | and |A i ∩ A j | are independent binomial B(n, 1/2) and B(n, 1/4) respectively, by standard Chernoff bounds (see [12], chapter 5), it follows (by straightforward computations) that one can get such a family of cardinality at least e 2ε 2 n/75 . In fact this same construction gives super-polynomial sized families even if ε = n −1/2+δ for any fixed δ > 0.
Another interesting facet of the fractional intersection notion is the following extension of l-avoiding families [6,10]