Combinatorics of Intervals Preliminary Version

1.1. Intervals. We shall denote by I a finite set of closed intervals of the real line. Investigations of combinatorial properties of intervals started by Gy. Hajós (1957, [40]) T. Gallai ([26]) S. Benzer (1959, [8]) P. C. Gilmore and A. J. Hoffman (1964, [7]) D. R. Fulkerson and O. A. Gross (1965, [22]) C. G. Lekkerkerker and J. Ch. Boland (1962, [57]). It is traditional to display a set of intervals in a slided fashion (as in Figure 1) . Intervals can be interpreted in several ways, one of the most natural interpretation is to look at them as time-intervals. Application 1. Scheduling lectures. A university schedules lectures on a given day. The time spans of the lectures are given by intervals. Lectures with overlapping time spans must use different classrooms. At least how many classrooms are needed? How can one find an optimal schedule of the lectures? ([30].) Excercise 1. How many classrooms are needed to schedule lectures with time spans of Figure 1? Application 2. Storing chemicals. Chemicals must be stored in refrigerators. Each chemical must be stored at a constant temperature inside a given interval. At least how many refrigerators are needed? How can one find an optimal plan for storing them? ([30].) Exercise 2. How many refrigerators are needed to store the chemicals with temperature intervals of Figure 1?