Online and Quasi-online Colorings of Wedges and Intervals

We consider proper online colorings of hypergraphs defined by geometric regions. We prove that there is an online coloring method that colors N intervals of the real line using Θ(logN/k) colors such that for every point p, contained in at least k intervals, not all the intervals containing p have the same color. We also prove the corresponding result about online coloring quadrants in the plane that are parallel to a given fixed quadrant. These results contrast to recent results of the first and third author showing that in the quasi-online setting 12 colors are enough to color quadrants (independent of N and k). We also consider coloring intervals in the quasi-online setting. In all cases we present efficient coloring algorithms as well