Bounds on the range of random graph homomorphism into Z, and the maximal height diierence of the Gaussian random eld, are presented.

We show that certain canonical realizations of the complexes Hom(G,H) and Hom+(G,H) of (partial) graph homomorphisms studied by Babson and Kozlov are in fact instances of the polyhedral Cayley trick. For G a complete graph, we then characterize when a canonical projection of these complexes is itself again a complex, and exhibit several well-known objects that arise as cells or subcomplexes of ...

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Let Gn,k denote the Kneser graph whose vertices are the n-element subsets of a (2n + k)-element set and whose edges are the disjoint pairs. In this paper we prove that for any non-negative integer s there is no graph homomorphism from G4,2 to G4s+1,2s+1. This confirms a conjecture of Stahl in a special case.

Let Φ d,n be the fundamental group of the space of smooth projective hypersurfaces of degree d and dimension n and let ρ be its natural monodromy representation. Then the kernel of ρ is large for d ≥ 3 with the exception of the cases (d, n) = (3, 0), (3, 1). For these and for d < 3 the kernel is finite. A large group is one that admits a homomorphism to a semisimple Lie group of noncompact type...

Here CH0(X) denotes the Chow group of zero-cycles on X modulo rational equivalence and π̃ 1 (X) ab is the modified abelianized étale fundamental group, which classifies finite abelian étale coverings of X in which every real point splits completely (if X is defined over a totally imaginary number field, then this is just the usual abelianized étale fundamental group). By Lang [La], the homomorph...

We consider the question of the existence of homomorphisms between Gn,p and odd cycles when p = c/n, 1 < c ≤ 4. We show that for any positive integer l, there exists ε = ε(l) such that if c = 1 + ε then w.h.p. Gn,p has a homomorphism from Gn,p to C2l+1 so long as its odd-girth is at least 2l+1. On the other hand, we show that if c = 4 then w.h.p. there is no homomorphism from Gn,p to C5. Note t...

for fr$acute{mathbf{text{e}}}$chet algebras $(a, (p_n))$ and $(b, (q_n))$, a linear map $t:arightarrow b$ is textit{almost multiplicative} with respect to $(p_n)$ and $(q_n)$, if there exists $varepsilongeq 0$ such that $q_n(tab - ta tb)leq varepsilon p_n(a) p_n(b),$ for all $n in mathbb{n}$, $a, b in a$, and it is called textit{weakly almost multiplicative} with respect to $(p_n)$ and $(q_n)$,...