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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 assigned one of $k$ colours, such that distinct points $v,win P$ are assigned the same colour if and only if some other point in $P$ blocks $v$ and $w$. The focus of this paper is the conjecture that each $k$-blocked set has bounded size (as a function of $k$). Results in the literature imply that every 2-blocked set has at most 3 points, and every 3-blocked set has at most 6 points. We prove that every 4-blocked set has at most 12 points, and that this bound is tight. In fact, we characterise all sets ${n_1,n_2,n_3,n_4}$ such that some 4-blocked set has exactly $n_i$ points in the $i$-th colour class. Amongst other results, for infinitely many values of $k$, we construct $k$-blocked sets with $k^{1.79...}$ points.
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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.
Let $G$ be an edge-coloured graph. The minimum colour degree $delta^c(G)$ of $G$ is the largest integer $k$ such that, for every vertex $v$, there are at least $k$ distinct colours on edges incident to $v$. We say that $G$ is properly coloured if no two adjacent edges have the same colour. In this paper, we show that, for any $varepsilon >0$ and $n$ large, every edge-coloured graph $G$ with $delta^c(G) ge (1/2+varepsilon)n$ contains a properly coloured cycle of length at least $min{ n , lfloor 2 delta^c(G)/3 rfloor}$.
Given $n$ points in the plane, a emph{covering path} is a polygonal path that visits all the points. If no three points are collinear, every covering path requires at least $n/2$ segments, and $n-1$ straight line segments obviously suffice even if the covering path is required to be noncrossing. We show that every set of $n$ points in the plane admits a (possibly self-crossi ng) covering path consisting of $n/2 +O(n/log{n})$ straight line segments. If the path is required to be noncrossing, we prove that $(1-eps)n$ straight line segments suffice for a small constant $eps>0$, and we exhibit $n$-element point sets that require at least $5n/9 -O(1)$ segments in every such path. Further, the analogous question for noncrossing emph{covering trees} is considered and similar bounds are obtained. Finally, it is shown that computing a noncrossing covering path for $n$ points in the plane requires $Omega(n log{n})$ time in the worst case.
Given a finite set $A subseteq mathbb{R}^d$, points $a_1,a_2,dotsc,a_{ell} in A$ form an $ell$-hole in $A$ if they are the vertices of a convex polytope which contains no points of $A$ in its interior. We construct arbitrarily large point sets in general position in $mathbb{R}^d$ having no holes of size $O(4^ddlog d)$ or more. This improves the previously known upper bound of order $d^{d+o(d)}$ due to Valtr. The basic version of our construction uses a certain type of equidistributed point sets, originating from numerical analysis, known as $(t,m,s)$-nets or $(t,s)$-sequences, yielding a bound of $2^{7d}$. The better bound is obtained using a variant of $(t,m,s)$-nets, obeying a relaxed equidistribution condition.
Geometrical objects with integral sides have attracted mathematicians for ages. For example, the problem to prove or to disprove the existence of a perfect box, that is, a rectangular parallelepiped with all edges, face diagonals and space diagonals of integer lengths, remains open. More generally an integral point set $mathcal{P}$ is a set of $n$ points in the $m$-dimensional Euclidean space $mathbb{E}^m$ with pairwise integral distances where the largest occurring distance is called its diameter. From the combinatorial point of view there is a natural interest in the determination of the smallest possible diameter $d(m,n)$ for given parameters $m$ and $n$. We give some new upper bounds for the minimum diameter $d(m,n)$ and some exact values.
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