Subfamily Of Size Research Articles

Inclusion Matrices and the MDS Conjecture

Let $\mathbb{F}_{q}$ be a finite field of order $q$ with characteristic $p$. An arc is an ordered family of at least $k$ vectors in $\mathbb{F}_{q}^{k}$ in which every subfamily of size $k$ is a basis of $\mathbb{F}_{q}^{k}$. The MDS conjecture, which was posed by Segre in 1955, states that if $k \leq q$, then an arc in $\mathbb{F}_{q}^{k}$ has size at most $q+1$, unless $q$ is even and $k=3$ or $k=q-1$, in which case it has size at most $q+2$. We propose a conjecture which would imply that the MDS conjecture is true for almost all values of $k$ when $q$ is odd. We prove our conjecture in two cases and thus give simpler proofs of the MDS conjecture when $k \leq p$, and if $q$ is not prime, for $k \leq 2p-2$. To accomplish this, given an arc $G \subset \mathbb{F}_{q}^{k}$ and a nonnegative integer $n$, we construct a matrix $M_{G}^{\uparrow n}$, which is related to an inclusion matrix, a well-studied object in combinatorics. Our main results relate algebraic properties of the matrix $M_{G}^{\uparrow n}$ to properties of the arc $G$ and may provide new tools in the computational classification of large arcs.

Regular systems of paths and families of convex sets in convex position

In this paper we show that every sufficiently large family of convex bodies in the plane has a large subfamily in convex position provided that the number of common tangents of each pair of bodies is bounded and every subfamily of size five is in convex position. (If each pair of bodies have at most two common tangents it is enough to assume that every triple is in convex position, and likewise, if each pair of bodies have at most four common tangents it is enough to assume that every quadruple is in convex position.) This confirms a conjecture of Pach and Toth, and generalizes a theorem of Bisztriczky and Fejes Toth. Our results on families of convex bodies are consequences of more general Ramsey-type results about the crossing patterns of systems of graphs of continuous functions $f:[0,1] \to \mathbb{R}$. On our way towards proving the Pach-Toth conjecture we obtain a combinatorial characterization of such systems of graphs in which all subsystems of equal size induce equivalent crossing patterns. These highly organized structures are what we call regular systems of paths and they are natural generalizations of the notions of cups and caps from the famous theorem of Erdos and Szekeres. The characterization of regular systems is combinatorial and introduces some auxiliary structures which may be of independent interest.

Analogues of the central point theorem for families with d-intersection property in ℝ d

In this paper we consider families of compact convex sets in ℝd such that any subfamily of size at most d has a nonempty intersection. We prove some analogues of the central point theorem and Tverberg’s theorem for such families.

NEW RESULTS FOR T ( k )‐FAMILIES IN THE PLANE

A line that intersects every member of a finite family F of convex sets in the plane is called a common transversal to F . In this paper we study some basic properties of T ( k )-families: finite families of convex sets in the plane in which every subfamily of size at most k admits a common transversal. It is known that a T ( k )-family admits a partial transversal of size α ∣ F ∣ for some constant α ( k ) which is independent of F . Here it will be shown that (2/( k ( k −1))) 1/( k −2) ≤ α ( k )≤(( k −2)/( k −1)), which are the best bounds to date.

On large intersecting subfamilies of uniform setfamilies

AbstractIn an earlier work R. A. Duke and V. Rödl, The Erdős–Ko–Rado theorem for small families, J Combin Theory Ser A 65(2) (1994), 246–251 it was shown that for t a fixed positive integer and κ a real constant, 0 < κ < 1/2, if n is sufficiently large each family 𝒜 of ⌊κn⌋‐element subsets of [n] of size N (linear in n) contains a t‐intersecting subfamily of size at least (1 − o(1))κN. Here we consider the case when t, the intersection size, is no longer bounded, specifically t = ⌊τn⌋ for 0 < τ < κ. We show that for sufficiently large n and N each family of this type contains an r‐wise t‐intersecting subfamily of size at least N1−δ, and that, apart for the size of δ, this result is the best possible. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 2003

A Helly Type Conjecture

A family of sets is Π n , or n -pierceable, if there exists a set of n points such that each member of the family contains at least one of them. It is Π k n if every subfamily of size k or less is Π n . Helly's theorem is one of the fundamental results in Combinatorial Geometry. It asserts, in the special case of finite families of convex sets in the plane, that Π 3 1 implies Π 1 . However, there is no k such that Π k 2 implies 2 -pierceability for all finite families of convex sets in the plane. It is therefore natural to propose the following: