On hierarchically closed fractional intersecting families

For a set $L$ of positive proper fractions and a positive integer $r \geq 2$, a fractional $r$-closed $L$-intersecting family is a collection $\mathcal{F} \subset \mathcal{P}([n])$ with the property that for any $2 \leq t \leq r$ and $A_1, \dotsc, A_t \in \mathcal{F}$ there exists $\theta \in L$ such that $\lvert A_1 \cap \dotsb \cap A_t \rvert \in \{ \theta \lvert A_1 \rvert, \dotsc, \theta \lvert A_t \rvert\}$. In this paper we show that for $r \geq 3$ and $L = \{\theta\}$ any fractional $r$-closed $\theta$-intersecting family has size at most linear in $n$, and this is best possible up to a constant factor. We also show that in the case $\theta = 1/2$ we have a tight upper bound of $\lfloor \frac{3n}{2} \rfloor - 2$ and that a maximal $r$-closed $(1/2)$-intersecting family is determined uniquely up to isomorphism.Comment: 18 pages, 0 figure

Bounded fractional intersecting families are linear in size

Using the sunflower method, we show that if $\theta \in (0,1) \cap \mathbb{Q}$ and $\mathcal{F}$ is a $O(n^{1/3})$-bounded $\theta$-intersecting family over $[n]$, then $\lvert \mathcal{F} \rvert = O(n)$, and that if $\mathcal{F}$ is $o(n^{1/3})$-bounded, then $\lvert \mathcal{F} \rvert \leq (\frac{3}{2} + o(1))n$. This partially solves a conjecture raised in (Balachandran et al., Electron J. Combin. 26 (2019), #P2.40) that any $\theta$-intersecting family over $[n]$ has size at most linear in $n$, in the regime where we have no very large sets.Comment: 7 page

Modular and fractional L-intersecting families of vector spaces

This paper is divided into two logical parts. In the first part of this paper, we prove the following theorem which is the q-analogue of a generalized modular Ray-Chaudhuri-Wilson Theorem shown in [Alon, Babai, Suzuki, J. Combin. Theory Series A, 1991]. It is also a generalization of the main theorem in [Frankl and Graham, European J. Combin. 1985] under certain circumstances.Let V be a vector space of dimension n over a finite field of size q. Let K = {k1,…,kr}, L = {μ1,…,μs} be two disjoint subsets of {0,1,…,b-1} with k1<….< kr. Let F = {V1,V2,…,Vm} be a family of subspaces of V such that (a) for every i ∈ [m], dim(Vi) mod b = kt, for some kt ∈ K, and (b) for every distinct i,j ∈ [m], dim(Vi ∩ Vj)mod b = μt, for some μt ∈ L. Moreover, it is given that neither of the following two conditions hold:(i) q + 1 is a power of 2, and b = 2 (ii) q = 2, b = 6. Then, (formula presented) otherwise, where (formula presented) In the second part of this paper, we prove q-analogues of results on a recent notion called fractional L-intersecting family of sets for families of subspaces of a given vector space over a finite field of size q. We use the above theorem to obtain a general upper bound to the cardinality of such families. We give an improvement to this general upper bound in certain special cases. © The authors