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(log N/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.
展开▼
机译：我们考虑了由几何区域定义的超图的正确在线着色。我们证明有一种在线着色方法，使用θ（log n / k）颜色为真实线的n个间隔颜色，使得对于每个点p，其中包含至少k间隔，并非所有包含p的间隔都具有相同的颜色。我们还证明了对与给定固定象限平行的平面中的在线着色象限的相应结果。这些结果与第一和第三作者的最近结果形成鲜明对比，显示在准联机设置中，12种颜色足以色彩象限（独立于N和K）。我们还考虑了准在线环境中的着色间隔。在所有情况下，我们也提出了高效的着色算法。
展开▼
