ﻻ يوجد ملخص باللغة العربية
We prove that a minimal $t$-fold blocking set in a finite projective plane of order $n$ has cardinality at most [frac{1}{2} nsqrt{4tn - (3t + 1)(t - 1)} + frac{1}{2} (t - 1)n + t.] This is the first general upper bound on the size of minimal $t$-fold blocking sets in finite projective planes and it generalizes the classical result of Bruen and Thas on minimal blocking sets. From the proof it directly follows that if equality occurs in this bound then every line intersects the blocking set $S$ in either $t$ points or $frac{1}{2}(sqrt{4tn - (3t + 1)(t - 1)} + t - 1) + 1$ points. We use this to show that for $n$ a prime power, equality can occur in our bound in exactly one of the following three cases: (a) $t = 1$, $n$ is a square and $S$ is a unital; (b) $t = n - sqrt{n}$, $n$ is a square and $S$ is the complement of a Baer subplane; (c) $t = n$ and $S$ is equal to the set of all points except one. For a square prime power $q$ and $t leq sqrt{q} + 1$, we give a construction of a minimal $t$-fold blocking set $S$ in $mathrm{PG}(2,q)$ with $|S| = qsqrt{q} + 1 + (t - 1)(q - sqrt{q} + 1)$. Furthermore, we obtain an upper bound on the size of minimal blocking sets in symmetric $2$-designs and use it to give new proofs of other known results regarding tangency sets in higher dimensional finite projective spaces. We also discuss further generalizations of our bound. In our proofs we use an incidence bound on combinatorial designs which follows from applying the expander mixing lemma to the incidence graph of these designs.
This paper studies problems related to visibility among points in the plane. A point $x$ emph{blocks} two points $v$ and $w$ if $x$ is in the interior of the line segment $bar{vw}$. A set of points $P$ is emph{$k$-blocked} if each point in $P$ is ass
For an integer $qge 2$, a graph $G$ is called $q$-Ramsey for a graph $H$ if every $q$-colouring of the edges of $G$ contains a monochromatic copy of $H$. If $G$ is $q$-Ramsey for $H$, yet no proper subgraph of $G$ has this property then $G$ is called
We show that the horocycle flows of open tight hyperbolic surfaces do not admit minimal sets.
Let $Lsubset mathbb{Z}^n$ be a lattice and $I_L=langle x^{bf u}-x^{bf v}: {bf u}-{bf v}in Lrangle$ be the corresponding lattice ideal in $Bbbk[x_1,ldots, x_n]$, where $Bbbk$ is a field. In this paper we describe minimal binomial generating sets of $I
This paper addresses the problem of finding minimum forcing sets in origami. The origami material folds flat along straight lines called creases that can be labeled as mountains or valleys. A forcing set is a subset of creases that force all the othe