Another look at color primitivism

This article is on a precise kind of color primitivism, ‘ostensivism.’ This is the view that it is in the nature of the colors that they are phenomenal, non-reductive, structural, categorical properties. First, I differentiate ostensivism from other precise forms of primitivism. Next, I examine the core belief ‘Revelation,’ and propose a revised version, which, unlike standard statements, is compatible with a yet unstated but plausible core belief: roughly, that there are interesting things to be discovered about the nature of the colors. Finally, I show that ostensivism is the only view on color that can accommodate both proposed core beliefs

Super-polylogarithmic hypergraph coloring hardness via low-degree long codes

We prove improved inapproximability results for hypergraph coloring using the low-degree polynomial code (aka, the 'short code' of Barak et. al. [FOCS 2012]) and the techniques proposed by Dinur and Guruswami [FOCS 2013] to incorporate this code for inapproximability results. In particular, we prove quasi-NP-hardness of the following problems on $n$-vertex hyper-graphs: * Coloring a 2-colorable 8-uniform hypergraph with $2^{2^{\Omega(\sqrt{\log\log n})}}$ colors. * Coloring a 4-colorable 4-uniform hypergraph with $2^{2^{\Omega(\sqrt{\log\log n})}}$ colors. * Coloring a 3-colorable 3-uniform hypergraph with $(\log n)^{\Omega(1/\log\log\log n)}$ colors. In each of these cases, the hardness results obtained are (at least) exponentially stronger than what was previously known for the respective cases. In fact, prior to this result, polylog n colors was the strongest quantitative bound on the number of colors ruled out by inapproximability results for O(1)-colorable hypergraphs. The fundamental bottleneck in obtaining coloring inapproximability results using the low- degree long code was a multipartite structural restriction in the PCP construction of Dinur-Guruswami. We are able to get around this restriction by simulating the multipartite structure implicitly by querying just one partition (albeit requiring 8 queries), which yields our result for 2-colorable 8-uniform hypergraphs. The result for 4-colorable 4-uniform hypergraphs is obtained via a 'query doubling' method. For 3-colorable 3-uniform hypergraphs, we exploit the ternary domain to design a test with an additive (as opposed to multiplicative) noise function, and analyze its efficacy in killing high weight Fourier coefficients via the pseudorandom properties of an associated quadratic form.Comment: 25 page

Hunting Galaxies to (and for) Extinction

In studies of star-forming regions, near-infrared excess (NIRX) sources--objects with intrinsic colors redder than normal stars--constitute both signal (young stars) and noise (e.g. background galaxies). We hunt down (identify) galaxies using near-infrared observations in the Perseus star-forming region by combining structural information, colors, and number density estimates. Galaxies at moderate redshifts (z = 0.1 - 0.5) have colors similar to young stellar objects (YSOs) at both near- and mid-infrared (e.g. Spitzer) wavelengths, which limits our ability to identify YSOs from colors alone. Structural information from high-quality near-infrared observations allows us to better separate YSOs from galaxies, rejecting 2/5 of the YSO candidates identified from Spitzer observations of our regions and potentially extending the YSO luminosity function below K of 15 magnitudes where galaxy contamination dominates. Once they are identified we use galaxies as valuable extra signal for making extinction maps of molecular clouds. Our new iterative procedure: the Galaxies Near Infrared Color Excess method Revisited (GNICER), uses the mean colors of galaxies as a function of magnitude to include them in extinction maps in an unbiased way. GNICER increases the number of background sources used to probe the structure of a cloud, decreasing the noise and increasing the resolution of extinction maps made far from the galactic plane.Comment: 16 pages and 16 figures. Accepted for publication in ApJ. Full resolution version at http://www.cfa.harvard.edu/COMPLETE/papers/Foster_HuntingGalaxies.pd

Approximation Algorithms for Partially Colorable Graphs

Graph coloring problems are a central topic of study in the theory of algorithms. We study the problem of partially coloring partially colorable graphs. For alpha = alpha |V| such that the graph induced on S is k-colorable. Partial k-colorability is a more robust structural property of a graph than k-colorability. For graphs that arise in practice, partial k-colorability might be a better notion to use than k-colorability, since data arising in practice often contains various forms of noise. We give a polynomial time algorithm that takes as input a (1 - epsilon)-partially 3-colorable graph G and a constant gamma in [epsilon, 1/10], and colors a (1 - epsilon/gamma) fraction of the vertices using O~(n^{0.25 + O(gamma^{1/2})}) colors. We also study natural semi-random families of instances of partially 3-colorable graphs and partially 2-colorable graphs, and give stronger bi-criteria approximation guarantees for these family of instances