Ramsey Theory Problems over the Integers: Avoiding Generalized Progressions

Two well studied Ramsey-theoretic problems consider subsets of the natural numbers which either contain no three elements in arithmetic progression, or in geometric progression. We study generalizations of this problem, by varying the kinds of progressions to be avoided and the metrics used to evaluate the density of the resulting subsets. One can view a 3-term arithmetic progression as a sequence $x, f_n(x), f_n(f_n(x))$, where $f_n(x) = x + n$, $n$ a nonzero integer. Thus avoiding three-term arithmetic progressions is equivalent to containing no three elements of the form $x, f_n(x), f_n(f_n(x))$ with $f_n \in\mathcal{F}_{\rm t}$, the set of integer translations. One can similarly construct related progressions using different families of functions. We investigate several such families, including geometric progressions ($f_n(x) = nx$ with $n > 1$ a natural number) and exponential progressions ($f_n(x) = x^n$). Progression-free sets are often constructed "greedily," including every number so long as it is not in progression with any of the previous elements. Rankin characterized the greedy geometric-progression-free set in terms of the greedy arithmetic set. We characterize the greedy exponential set and prove that it has asymptotic density 1, and then discuss how the optimality of the greedy set depends on the family of functions used to define progressions. Traditionally, the size of a progression-free set is measured using the (upper) asymptotic density, however we consider several different notions of density, including the uniform and exponential densities.Comment: Version 1.0, 13 page

Seasonal Variation in Nymphal Blacklegged Tick Abundance in Southern New England Forests

In the northeastern United States, risk of human exposure to tick transmitted disease is primarily a function of the abundance of the blacklegged tick, Ixodes scapularis Say. We assessed seasonal variability in the abundance of nymphal stage I. scapularis over 13 yr, collected from several forested areas throughout Rhode Island. Specifically, we examined intraseasonal differences by using two temporally distinct tick collections made during the peak nymphal tick season. Intraseasonal factors significantly impacted tick abundance, with the June tick rate (mean = 40.42, SD = 14.79) significantly more abundant than the July tick rate (mean = 27.64, SD = 15.47). The greater variability in July (coefficient of variation: June, 36.61%; July, 55.95%) lead us to conclude June tick rates are relatively stable from year to year, whereas July tick rates contribute more to intraseasonal and yearly variation

On the Progenitor System of the Type Iax Supernova 2014dt in M61

We present pre-explosion and post-explosion Hubble Space Telescope images of the Type Iax supernova (SN Iax) 2014dt in M61. After astrometrically aligning these images, we do not detect any stellar sources at the position of the SN in the pre-explosion images to relatively deep limits (3 sigma limits of M_F438W > -5.0 mag and M_F814W > -5.9 mag). These limits are similar to the luminosity of SN 2012Z's progenitor system (M_F435W = -5.43 +/- 0.15 and M_F814W = -5.24 +/- 0.16 mag), the only probable detected progenitor system in pre-explosion images of a SN Iax, and indeed, of any white dwarf supernova. SN 2014dt is consistent with having a C/O white-dwarf primary/helium-star companion progenitor system, as was suggested for SN 2012Z, although perhaps with a slightly smaller or hotter donor. The data are also consistent with SN 2014dt having a low-mass red giant or main-sequence star companion. The data rule out main-sequence stars with M_init > 16 M_sun and most evolved stars with M_init > 8 M_sun as being the progenitor of SN 2014dt. Hot Wolf-Rayet stars are also allowed, but the lack of nearby bright sources makes this scenario unlikely. Because of its proximity (D = 12 Mpc), SN 2014dt is ideal for long-term monitoring, where images in ~2 years may detect the companion star or the luminous bound remnant of the progenitor white dwarf.Comment: 5 pages, 3 figures, submitted to ApJ