REU 2007 · Transfinite Combinatorics · Lecture 1
REU 2007 · Transfinite Combinatorics · Lecture 1 Instructor: L´aszl´oBabai Scribe: Damir Dzhafarov July 23, 2007. Partially revised by instructor. Last updated July 26, 12:00 AM 1.1 Transfinite Combinatorics and Toy Problems Lamp and switches. Imagine a lamp which is always on, and which assumes one of three different colors. The particular color of the lamp at a given time is completely determined by the configuration, at that time, of a row of three-way switches (i.e., switches that can each occupy one of three different states). If there are n switches, any such configuration can be represented by an n-tuple (x1, . , xn), where xi ∈ {0, 1, 2} for all i, that is, by an element of {0, 1, 2}n. If we let the set of possible colors of the lamp be {0, 1, 2}, we can thus regard the lamp’s color as a function of the configurations of the switches, i.e., as a function f : {0, 1, 2}n → {0, 1, 2}. The system of the lamp and switches is subject to a single rule, that if the position of each switch be changed, the color of the lamp changes as well. In other words, if (x1, . , xn) 0 0 0 and (x1, . , xn) are configurations of switches with xi 6= xi for all i, then f(x1, . , xn) 6= 0 0 f(x1, . , xn). Exercise 1.1.1. Prove that if there are only finitely many switches, then there must exist a dictator switch, meaning one which makes all other switches redundant in that all of its positions induce different colors of the lamp, and each of its positions always induces the same color, regardless of the configuration of the other switches.
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