A Bichromatic Incidence Bound and an Application

We prove a new, tight upper bound on the number of incidences between points and hyperplanes in Euclidean d-space. Given n points, of which k are colored red, there are O_d(m^{2/3}k^{2/3}n^{(d-2)/3} + kn^{d-2} + m) incidences between the k red points and m hyperplanes spanned by all n points provided that m = \Omega(n^{d-2}). For the monochromatic case k = n, this was proved by Agarwal and Aronov. We use this incidence bound to prove that a set of n points, no more than n-k of which lie on any plane or two lines, spans \Omega(nk^2) planes. We also provide an infinite family of counterexamples to a conjecture of Purdy's on the number of hyperplanes spanned by a set of points in dimensions higher than 3, and present new conjectures not subject to the counterexample.Comment: 12 page

Helly numbers of Algebraic Subsets of Rd\mathbb R^d

We study $S$-convex sets, which are the geometric objects obtained as the intersection of the usual convex sets in $\mathbb R^d$ with a proper subset $S\subset \mathbb R^d$. We contribute new results about their $S$-Helly numbers. We extend prior work for $S=\mathbb R^d$, $\mathbb Z^d$, and $\mathbb Z^{d-k}\times\mathbb R^k$; we give sharp bounds on the $S$-Helly numbers in several new cases. We considered the situation for low-dimensional $S$ and for sets $S$ that have some algebraic structure, in particular when $S$ is an arbitrary subgroup of $\mathbb R^d$ or when $S$ is the difference between a lattice and some of its sublattices. By abstracting the ingredients of Lov\'asz method we obtain colorful versions of many monochromatic Helly-type results, including several colorful versions of our own results.Comment: 13 pages, 3 figures. This paper is a revised version of what was originally the first half of arXiv:1504.00076v

Quantitative Transversal Theorems in the Plane

Hadwiger's theorem is a variant of Helly-type theorems involving common transversals to families of convex sets instead of common intersections. In this paper, we obtain a quantitative version of Hadwiger's theorem on the plane: given an ordered family of pairwise disjoint and compact convex sets in $\mathbb{R}^2$ and any real-valued monotone function on convex subsets of $\mathbb{R}^2,$ if every three sets have a common transversal, respecting the order, such that the intersection of the sets with each half-plane defined by the transversal are valued at least (or at most) some constant $\alpha,$ then the entire family has a common transversal with the same property. Unlike previous generalizations of Hadwiger's theorem, we prove that disjointness is necessary for the quantitative case. We also prove colorful versions of our results

SparsePak: A Formatted Fiber Field-Unit for The WIYN Telescope Bench Spectrograph. II. On-Sky Performance

We present a performance analysis of SparsePak and the WIYN Bench Spectrograph for precision studies of stellar and ionized gas kinematics of external galaxies. We focus on spectrograph configurations with echelle and low-order gratings yielding spectral resolutions of ~10000 between 500-900nm. These configurations are of general relevance to the spectrograph performance. Benchmarks include spectral resolution, sampling, vignetting, scattered light, and an estimate of the system absolute throughput. Comparisons are made to other, existing, fiber feeds on the WIYN Bench Spectrograph. Vignetting and relative throughput are found to agree with a geometric model of the optical system. An aperture-correction protocol for spectrophotometric standard-star calibrations has been established using independent WIYN imaging data and the unique capabilities of the SparsePak fiber array. The WIYN point-spread-function is well-fit by a Moffat profile with a constant power-law outer slope of index -4.4. We use SparsePak commissioning data to debunk a long-standing myth concerning sky-subtraction with fibers: By properly treating the multi-fiber data as a ``long-slit'' it is possible to achieve precision sky subtraction with a signal-to-noise performance as good or better than conventional long-slit spectroscopy. No beam-switching is required, and hence the method is efficient. Finally, we give several examples of science measurements which SparsePak now makes routine. These include H$\alpha$ velocity fields of low surface-brightness disks, gas and stellar velocity-fields of nearly face-on disks, and stellar absorption-line profiles of galaxy disks at spectral resolutions of ~24,000.Comment: To appear in ApJSupp (Feb 2005); 19 pages text; 7 tables; 27 figures (embedded); high-resolution version at http://www.astro.wisc.edu/~mab/publications/spkII_pre.pd

The Incidence Hopf Algebra of Graphs

This is the published version, also available here: http://dx.doi.org/10.1137/110820075.The graph algebra is a commutative, cocommutative, graded, connected incidence Hopf algebra, whose basis elements correspond to finite graphs, and whose Hopf product and coproduct admit simple combinatorial descriptions. We give a new formula for the antipode in the graph algebra in terms of acyclic orientations; our formula contains many fewer terms than Takeuchi's and Schmitt's more general formulas for the antipode in an incidence Hopf algebra. Applications include several formulas (some old and some new) for evaluations of the Tutte polynomial

Discrete Geometry

A number of important recent developments in various branches of discrete geometry were presented at the workshop. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics or algebraic geometry. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry

Bounding Radon Number via Betti Numbers

We prove general topological Radon-type theorems for sets in ?^d, smooth real manifolds or finite dimensional simplicial complexes. Combined with a recent result of Holmsen and Lee, it gives fractional Helly theorem, and consequently the existence of weak ?-nets as well as a (p,q)-theorem. More precisely: Let X be either ?^d, smooth real d-manifold, or a finite d-dimensional simplicial complex. Then if F is a finite, intersection-closed family of sets in X such that the ith reduced Betti number (with ?? coefficients) of any set in F is at most b for every non-negative integer i less or equal to k, then the Radon number of F is bounded in terms of b and X. Here k is the smallest integer larger or equal to d/2 - 1 if X = ?^d; k=d-1 if X is a smooth real d-manifold and not a surface, k=0 if X is a surface and k=d if X is a d-dimensional simplicial complex. Using the recent result of the author and Kalai, we manage to prove the following optimal bound on fractional Helly number for families of open sets in a surface: Let F be a finite family of open sets in a surface S such that the intersection of any subfamily of F is either empty, or path-connected. Then the fractional Helly number of F is at most three. This also settles a conjecture of Holmsen, Kim, and Lee about an existence of a (p,q)-theorem for open subsets of a surface

Bounding Radon's number via Betti numbers

We prove general topological Radon type theorems for sets in $\mathbb R^d$, smooth real manifolds or finite dimensional simplicial complexes. Combined with a recent result of Holmsen and Lee, it gives fractional Helly and colorful Helly theorems, and consequently an existence of weak $\varepsilon$-nets as well as a $(p,q)$-theorem. More precisely: Let $X$ be either $\mathbb R^d$, smooth real $d$-manifold, or a finite $d$-dimensional simplicial complex. Then if $\mathcal F$ is a finite family of sets in $X$ such that $\widetilde\beta_i(\bigcap \mathcal G; \mathbb Z_2)$ is at most $b$ for all $i=0,1,\ldots, k$ and $\mathcal G\subseteq \mathcal F$, then the Radon's number of $\mathcal F$ is bounded in terms of $b$ and $X$. Here $k=\left\lceil\frac{d}{2}\right\rceil-1$ if $X=\mathbb R^d$; $k=d-1$ if $X$ is a smooth real $d$-manifold and not a surface, $k=0$ if $X$ is a surface and $k=d$ if $X$ is a $d$-dimensional simplicial complex. Using the recent result of the author and Kalai, we manage to prove the following optimal bound on fractional Helly number for families of open sets in a surface: Let $\mathcal F$ be a finite family of open sets in a surface $S$ such that for every $\mathcal G\subseteq \mathcal F$, $\bigcap \mathcal G$ is either empty, or path-connected. Then the fractional Helly number of $\mathcal F$ is at most three. This also settles a conjecture of Holmsen, Kim, and Lee about an existence of a $(p,q)$-theorem for open subsets of a surface.Comment: 11 pages, 2 figure

Rainbow and monochromatic circuits and cocircuits in binary matroids

Given a matroid together with a coloring of its ground set, a subset of its elements is called rainbow colored if no two of its elements have the same color. We show that if a binary matroid of rank $r$ is colored with exactly $r$ colors, then $M$ either contains a rainbow colored circuit or a monochromatic cut. As the class of binary matroids is closed under taking duals, this immediately implies that $M$ either contains a rainbow colored cut or a monochromatic circuit as well. As a byproduct, we give a characterization of binary matroids in terms of reductions to partition matroids. Motivated by a conjecture of B\'erczi et al., we also analyze the relation between the covering number of a binary matroid and the maximum number of colors or the maximum size of a color class in any of its rainbow circuit-free colorings. For simple graphic matroids, we show that there exists a rainbow circuit-free coloring that uses each color at most twice only if the graph is $(2,3)$-sparse, that is, it is independent in the $2$-dimensional rigidity matroid. Furthermore, we give a complete characterization of minimally rigid graphs admitting such a coloring.Comment: 15 pages, 1 figur