The number of k-intersections of an intersecting family of r-sets

The Erdos-Ko-Rado theorem tells us how large an intersecting family of r-sets from an n-set can be, while results due to Lovasz and Tuza give bounds on the number of singletons that can occur as pairwise intersections of sets from such a family. We consider a natural generalization of these problems. Given an intersecting family of r-sets from an n-set and 1\leq k \leq r, how many k-sets can occur as pairwise intersections of sets from the family? For k=r and k=1 this reduces to the problems described above. We answer this question exactly for all values of k and r, when n is sufficiently large. We also characterize the extremal families.Comment: 10 pages, 1 figur

Set Systems with No Singleton Intersection

Let $\mathcal{F}$ be a $k$-uniform set system defined on a ground set of size $n$ with no singleton intersection; i.e., no pair $A,B\in\mathcal{F}$ has $|A\cap B|=1$. Frankl showed that $|\mathcal{F}|\leq\binom{n-2}{k-2}$ for $k\geq4$ and $n$ sufficiently large, confirming a conjecture of Erdős and Sós. We determine the maximum size of $\mathcal{F}$ for $k=4$ and all $n$, and also establish a stability result for general $k$, showing that any $\mathcal{F}$ with size asymptotic to that of the best construction must be structurally similar to it

Set Systems with Restricted Cross-Intersections and the Minimum Rank of Inclusion Matrices

A set system is L-intersecting if any pairwise intersection size lies in L, where L is some set of s nonnegative integers. The celebrated Frankl-Ray-Chaudhuri-Wilson theorems give tight bounds on the size of an L-intersecting set system on a ground set of size n. Such a system contains at most $\binom{n}{s}$ sets if it is uniform and at most $\sum_{i=0}^s \binom{n}{i}$ sets if it is nonuniform. They also prove modular versions of these results. We consider the following extension of these problems. Call the set systems $\mathcal{A}_1,\ldots,\mathcal{A}_k$ {\em L-cross-intersecting} if for every pair of distinct sets A,B with $A \in \mathcal{A}_i$ and $B \in \mathcal{A}_j$ for some $i \neq j$ the intersection size $|A \cap B|$ lies in $L$. For any k and for n > n 0 (s) we give tight bounds on the maximum of $\sum_{i=1}^k |\mathcal{A}_i|$. It is at most $\max\, \{k\binom{n}{s}, \binom{n}{\lfloor n/2 \rfloor}\}$ if the systems are uniform and at most $\max\, \{k \sum_{i=0}^s \binom{n}{i} , (k-1) \sum_{i=0}^{s-1} \binom{n}{i} + 2^n\}$ if they are nonuniform. We also obtain modular versions of these results. Our proofs use tools from linear algebra together with some combinatorial ideas. A key ingredient is a tight lower bound for the rank of the inclusion matrix of a set system. The s*-inclusion matrix of a set system $\mathcal{A}$ on [n] is a matrix M with rows indexed by $\mathcal{A}$ and columns by the subsets of [n] of size at most s, where if $A \in \mathcal{A}$ and $B \subset [n]$ with $|B| \leq s$, we define M AB to be 1 if $B \subset A$ and 0 otherwise. Our bound generalizes the well-known result that if $|\mathcal{A}| < 2^{s+1}$, then M has full rank $|\mathcal{A}|$. In a combinatorial setting this fact was proved by Frankl and Pach in the study of null t-designs; it can also be viewed as determining the minimum distance of the Reed-Muller codes

Coloring intersection graphs of arc-connected sets in the plane

A family of sets in the plane is simple if the intersection of its any subfamily is arc-connected, and it is pierced by a line $L$ if the intersection of its any member with $L$ is a nonempty segment. It is proved that the intersection graphs of simple families of compact arc-connected sets in the plane pierced by a common line have chromatic number bounded by a function of their clique number.Comment: Minor changes + some additional references not included in the journal versio

A formula for the number of tilings of an octagon by rhombi

We propose the first algebraic determinantal formula to enumerate tilings of a centro-symmetric octagon of any size by rhombi. This result uses the Gessel-Viennot technique and generalizes to any octagon a formula given by Elnitsky in a special case.Comment: New title. Minor improvements. To appear in Theoretical Computer Science, special issue on "Combinatorics of the Discrete Plane and Tilings

Almost-Fisher families

A classic theorem in combinatorial design theory is Fisher's inequality, which states that a family $\mathcal F$ of subsets of $[n]$ with all pairwise intersections of size $\lambda$ can have at most $n$ non-empty sets. One may weaken the condition by requiring that for every set in $\mathcal F$, all but at most $k$ of its pairwise intersections have size $\lambda$. We call such families $k$-almost $\lambda$-Fisher. Vu was the first to study the maximum size of such families, proving that for $k=1$ the largest family has $2n-2$ sets, and characterising when equality is attained. We substantially refine his result, showing how the size of the maximum family depends on $\lambda$. In particular we prove that for small $\lambda$ one essentially recovers Fisher's bound. We also solve the next open case of $k=2$ and obtain the first non-trivial upper bound for general $k$.Comment: 27 pages (incluiding one appendix