15 research outputs found
On the general position subset selection problem
Let be the maximum integer such that every set of points in
the plane with at most collinear contains a subset of points
with no three collinear. First we prove that if then
. Second we prove that if
then , which implies all previously known lower bounds on and
improves them when is not fixed. A more general problem is to consider
subsets with at most collinear points in a point set with at most
collinear. We also prove analogous results in this setting
General Position Subsets and Independent Hyperplanes in d-Space
Erd\H{o}s asked what is the maximum number such that every set of
points in the plane with no four on a line contains points in
general position. We consider variants of this question for -dimensional
point sets and generalize previously known bounds. In particular, we prove the
following two results for fixed :
- Every set of hyperplanes in contains a subset
of size at least , for some
constant , such that no cell of the arrangement of is bounded by
hyperplanes of only.
- Every set of points in , for some constant
, contains a subset of cohyperplanar points or points in
general position.
Two-dimensional versions of the above results were respectively proved by
Ackerman et al. [Electronic J. Combinatorics, 2014] and by Payne and Wood [SIAM
J. Discrete Math., 2013].Comment: 8 page
Covering Partial Cubes with Zones
A partial cube is a graph having an isometric embedding in a hypercube.
Partial cubes are characterized by a natural equivalence relation on the edges,
whose classes are called zones. The number of zones determines the minimal
dimension of a hypercube in which the graph can be embedded. We consider the
problem of covering the vertices of a partial cube with the minimum number of
zones. The problem admits several special cases, among which are the problem of
covering the cells of a line arrangement with a minimum number of lines, and
the problem of finding a minimum-size fibre in a bipartite poset. For several
such special cases, we give upper and lower bounds on the minimum size of a
covering by zones. We also consider the computational complexity of those
problems, and establish some hardness results
On Higher Dimensional Point Sets in General Position
A finite point set in ?^d is in general position if no d + 1 points lie on a common hyperplane. Let ?_d(N) be the largest integer such that any set of N points in ?^d with no d + 2 members on a common hyperplane, contains a subset of size ?_d(N) in general position. Using the method of hypergraph containers, Balogh and Solymosi showed that ??(N) < N^{5/6 + o(1)}. In this paper, we also use the container method to obtain new upper bounds for ?_d(N) when d ? 3. More precisely, we show that if d is odd, then ?_d(N) < N^{1/2 + 1/(2d) + o(1)}, and if d is even, we have ?_d(N) < N^{1/2 + 1/(d-1) + o(1)}.
We also study the classical problem of determining the maximum number a(d,k,n) of points selected from the grid [n]^d such that no k + 2 members lie on a k-flat. For fixed d and k, we show that a(d,k,n)? O(n^{d/{2?(k+2)/4?}(1- 1/{2?(k+2)/4?d+1})}), which improves the previously best known bound of O(n^{d/?(k + 2)/2?}) due to Lefmann when k+2 is congruent to 0 or 1 mod 4