The anti-van der Waerden number, denoted by aw([n], k), is the smallest r such that every exact r-coloring of [n] contains a rainbow k-term arithmetic progression. Butler et al. showed that dlog3 ne + 2 6 aw([n], 3) 6 dlog2 ne + 1, and conjectured that there exists a constant C such that aw([n], 3) 6 dlog3 ne + C. In this paper, we show this conjecture is true by determining aw([n], 3) for all ...

We investigate the question “To what extent can random reals be used as a tool to establish number theoretic facts?” Let 2-RAN be the principle that for every real X there is a real R which is 2-random relative to X . In Section 2, we observe that the arguments of Csima and Mileti [3] can be implemented in the base theory RCA0 and so RCA0 +2-RAN implies the Rainbow Ramsey Theorem. In Section 3,...

Global Rainbow Thermometry is presented. It is a new technique for measuring average size and temperature of spray droplets. For data inversion, a global rainbow pattern is employed, which is formed by constructive interference of laser-light scattering spherical droplets. The non-spherical droplets and liquid ligaments provide a uniform background and thus do not influence the interference pat...