On Infinite-finite Duality Pairs of Directed Graphs

Erdős, Péter and Tardif, C. and Tardos, Gábor (2013) On Infinite-finite Duality Pairs of Directed Graphs. ORDER-A JOURNAL ON THE THEORY OF ORDERED SETS AND ITS APPLICATIONS, 30 (3). pp. 807-819. ISSN 0167-8094

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Abstract

The {Mathematical expression} duality pairs play a crucial role in the theory of general relational structures and in Constraint Satisfaction Problems. The case where both sides are finite is fully characterized. The case where both sides are infinite seems to be very complex. It is also known that no finite-infinite duality pair is possible if we make the additional restriction that both classes are antichains. In this paper (which is the first one of a series) we start the detailed study of the infinite-finite case. Here we concentrate on directed graphs. We prove some elementary properties of the infinite-finite duality pairs, including lower and upper bounds on the size of {Mathematical expression}, and show that the elements of {Mathematical expression} must be equivalent to forests if {Mathematical expression} is an antichain. Then we construct instructive examples, where the elements of {Mathematical expression} are paths or trees. Note that the existence of infinite-finite antichain dualities was not previously known.

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