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44 KAM Mathematical Colloquium

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Prof. ERNST SPECKER

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ETH Zurich

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MODULAR COUNTING

March 5, 2002
Lecture Room K9, Charles University, Sokolovska 83, Praha 8
16:00 PM

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Abstract

Suppose that the cardinality *n* of a certain set is determined to
be 6942 in one paper and 7181 in another. What should we do? We can try
to decide whether *n* is even or odd. In order to find an answer to
such a problem, it is often advantageous to generalize it. If e.g. we want
to know whether the number of different equivalence relations on a set
of 1000 elements is even or odd, we study the function *B* where *B(n)*
is then the number of equivalence relations on a set of *n* elements.
It is easy to see that the *B(n)* mod 2 is periodic; the period being
3, the parity of *B(1000)* is equal to the parity of *B(1)*,
i.e. *B(1000)* is odd. Does a corresponding theorem hold for the function
which counts the number of 3-colourable graphs on a set of *n* elements?
The aim of the talk is to characterize structures for which the answer
is positive.