Bounding Helly numbers via Betti numbers

We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem. More precisely, we show that for any non-negative integers $b$ and $d$ there exists an integer $h(b,d)$ such that the following holds. If $\mathcal F$ is a finite family of subsets of $\mathbb R^d$ such that $\tilde\beta_i\left(\bigcap\mathcal G\right) \le b$ for any $\mathcal G \subsetneq \mathcal F$ and every $0 \le i \le \lceil d/2 \rceil-1$ then $\mathcal F$ has Helly number at most $h(b,d)$. Here $\tilde\beta_i$ denotes the reduced $\mathbb Z_2$-Betti numbers (with singular homology). These topological conditions are sharp: not controlling any of these $\lceil d/2 \rceil$ first Betti numbers allow for families with unbounded Helly number. Our proofs combine homological non-embeddability results with a Ramsey-based approach to build, given an arbitrary simplicial complex $K$, some well-behaved chain map $C_*(K) \to C_*(\mathbb R^d)$.Comment: 29 pages, 8 figure

Some remarks on barycentric-sum problems over cyclic groups

We derive some new results on the k-th barycentric Olson constants of abelian groups (mainly cyclic). This quantity, for a finite abelian (additive) group (G,+), is defined as the smallest integer l such that each subset A of G with at least l elements contains a subset with k elements {g_1, ..., g_k} satisfying g_1 + ... + g_k = k g_j for some 1 <= j <= k.Comment: to appear in European Journal of Combinatoric

Topological methods in zero-sum Ramsey theory

A cornerstone result of Erd\H os, Ginzburg, and Ziv (EGZ) states that any sequence of $2n-1$ elements in $\mathbb{Z}/n$ contains a zero-sum subsequence of length $n$. While algebraic techniques have predominated in deriving many deep generalizations of this theorem over the past sixty years, here we introduce topological approaches to zero-sum problems which have proven fruitful in other combinatorial contexts. Our main result (1) is a topological criterion for determining when any $\mathbb{Z}/n$-coloring of an $n$-uniform hypergraph contains a zero-sum hyperedge. In addition to applications for Kneser hypergraphs, for complete hypergraphs our methods recover Olson's generalization of the EGZ theorem for arbitrary finite groups. Furthermore, we (2) give a fractional generalization of the EGZ theorem with applications to balanced set families and (3) provide a constrained EGZ theorem which imposes combinatorial restrictions on zero-sum sequences in the original result.Comment: 18 page

The SDSS-III APOGEE Radial Velocity Survey Of M Dwarfs. I. Description Of The Survey And Science Goals

We are carrying out a large ancillary program with the Sloan Digital Sky Survey, SDSS-III, using the fiber-fed multi-object near-infrared APOGEE spectrograph, to obtain high-resolution H-band spectra of more than 1200 M dwarfs. These observations will be used to measure spectroscopic rotational velocities, radial velocities, physical stellar parameters, and variability of the target stars. Here, we describe the target selection for this survey, as well as results from the first year of scientific observations based on spectra that will be publicly available in the SDSS-III DR 10 data release. As part of this paper we present radial velocities and rotational velocities of over 200 M dwarfs, with a v sin i precision of similar to 2 km s(-1) a measurement floor at v sin i = 4 km s(-1). This survey significantly increases the number of M dwarfs studied for rotational velocities and radial velocity variability (at similar to 100-200 m s(-1)), and will inform and advance the target selection for planned radial velocity and photometric searches for low-mass exoplanets around M dwarfs, such as the Habitable Zone Planet Finder, CARMENES, and TESS. Multiple epochs of radial velocity observations enable us to identify short period binaries, and adaptive optics imaging of a subset of stars enables the detection of possible stellar companions at larger separations. The high-resolution APOGEE spectra, covering the entire H band, provide the opportunity to measure physical stellar parameters such as effective temperatures and metallicities for many of these stars. At the culmination of this survey, we will have obtained multi-epoch spectra and radial velocities for over 1400 stars spanning the spectral range M0-L0, providing the largest set of near-infrared M dwarf spectra at high resolution, and more than doubling the number of known spectroscopic a sin i values for M dwarfs. Furthermore, by modeling telluric lines to correct for small instrumental radial velocity shifts, we hope to achieve a relative velocity precision floor of 50 m s(-1) for bright M dwarfs. With three or more epochs, this precision is adequate to detect substellar companions, including giant planets with short orbital periods, and flag them for higher-cadence followup. We present preliminary, and promising, results of this telluric modeling technique in this paper.Center for Exoplanets and Habitable WorldsPennsylvania State UniversityEberly College of SciencePennsylvania Space Grant ConsortiumNSF AST 1006676, AST 1126413National Science FoundationNational Aeronautics and Space Administration NNX-08AE38A, NNX13AB03GAlfred P. Sloan FoundationU.S. Department of Energy Oce of ScienceUniversity of ArizonaBrazilian Participation GroupBrookhaven National LaboratoryUniversity of CambridgeCarnegie Mellon UniversityUniversity of FloridaFrench Participation GroupGerman Participation GroupHarvard UniversityInstituto de Astrosica de CanariasMichigan State/Notre Dame/JINA Participation GroupJohns Hopkins UniversityLawrence Berkeley National LaboratoryMax Planck Institute for AstrophysicsMax Planck Institute for Extraterrestrial PhysicsNew Mexico State UniversityNew York UniversityOhio State UniversityUniversity of PortsmouthPrinceton UniversitySpanish Participation GroupUniversity of TokyoUniversity of UtahVanderbilt UniversityUniversity of VirginiaUniversity of WashingtonYale UniversityMcDonald Observator

The SDSS-III APOGEE Radial Velocity Survey of M dwarfs I: Description of Survey and Science Goals

We are carrying out a large ancillary program with the SDSS-III, using the fiber-fed multi-object NIR APOGEE spectrograph, to obtain high-resolution H-band spectra of more than 1200 M dwarfs. These observations are used to measure spectroscopic rotational velocities, radial velocities, physical stellar parameters, and variability of the target stars. Here, we describe the target selection for this survey and results from the first year of scientific observations based on spectra that is publicly available in the SDSS-III DR10 data release. As part of this paper we present RVs and vsini of over 200 M dwarfs, with a vsini precision of ~2 km/s and a measurement floor at vsini = 4 km/s. This survey significantly increases the number of M dwarfs studied for vsini and RV variability (at ~100-200 m/s), and will advance the target selection for planned RV and photometric searches for low mass exoplanets around M dwarfs, such as HPF, CARMENES, and TESS. Multiple epochs of radial velocity observations enable us to identify short period binaries, and AO imaging of a subset of stars enables the detection of possible stellar companions at larger separations. The high-resolution H-band APOGEE spectra provide the opportunity to measure physical stellar parameters such as effective temperatures and metallicities for many of these stars. At the culmination of this survey, we will have obtained multi-epoch spectra and RVs for over 1400 stars spanning spectral types of M0-L0, providing the largest set of NIR M dwarf spectra at high resolution, and more than doubling the number of known spectroscopic vsini values for M dwarfs. Furthermore, by modeling telluric lines to correct for small instrumental radial velocity shifts, we hope to achieve a relative velocity precision floor of 50 m/s for bright M dwarfs. We present preliminary results of this telluric modeling technique in this paper.Comment: Submitted to Astronomical Journa

Colorful Borsuk--Ulam theorems and applications

We prove a colorful generalization of the Borsuk--Ulam theorem and derive colorful consequences from it, such as a colorful generalization of the ham sandwich theorem. Even in the uncolored case this specializes to a strengthening of the ham sandwich theorem, which given an additional condition, contains a result of B\'{a}r\'{a}ny, Hubard, and Jer\'{o}nimo on well-separated measures as a special case. We prove a colorful generalization of Fan's antipodal sphere covering theorem, we derive a short proof of Gale's colorful KKM theorem, and we prove a colorful generalization of Brouwer's fixed point theorem. Our results also provide an alternative between Radon-type intersection results and KKM-type covering results. Finally, we prove colorful Borsuk--Ulam theorems for higher symmetry.Comment: 15 page