On homomorphisms from the Hamming cube to Z |
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Authors: | David Galvin |
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Institution: | (1) Microsoft Research, One Microsoft Way, 98052 Redmond, WA, USA |
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Abstract: | WriteF for the set of homomorphisms from {0, 1}
d
toZ which send0 to 0 (think of members ofF as labellings of {0, 1}
d
in which adjacent strings get labels differing by exactly 1), andF
1 for those which take on exactlyi values. We give asymptotic formulae for |F| and |F|.
In particular, we show that the probability that a uniformly chosen memberf ofF takes more than five values tends to 0 asd→∞. This settles a conjecture of J. Kahn. Previously, Kahn had shown that there is a constantb such thatf a.s. takes at mostb values. This in turn verified a conjecture of I. Benjaminiet al., that for eacht>0,f a.s. takes at mosttd values.
Determining |F| is equivalent both to counting the number of rank functions on the Boolean lattice 2d] (functionsf: 2d]→N satisfyingf(
) andf(A)≤f(A∪x)≤f(A)+1 for allA∈2d] andx∈d]) and to counting the number of proper 3-colourings of the discrete cube (i.e., the number of homomorphisms from {0, 1}
d
toK
3, the complete graph on 3 vertices).
Our proof uses the main lemma from Kahn’s proof of constant range, together with some combinatorial approximation techniques
introduced by A. Sapozhenko.
Research supported by a Graduate School Fellowship from Rutgers University. |
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Keywords: | |
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