The asymptotics of 2-colour Ramsey numbers of loose and tight cycles in 3-uniform hypergraphs were recently determined [16, 17]. We address the same problem for Berge cycles and for 3 colours. Our main result is that the 3-colour Ramsey number of a 3-uniform Berge cycle of length n is asymptotic to . The result is proved with the Regularity Lemma via the existence of a monochromatic connected matching covering asymptotically 4n/5 vertices in the multicoloured 2-shadow graph induced by the colouring of Kn(3).

Let I be an independent set drawn from the discrete d-dimensional hypercube Qd = {0, 1}d according to the hard-core distribution with parameter λ > 0 (that is, the distribution in which each independent set I is chosen with probability proportional to λ|I|). We show a sharp transition around λ = 1 in the appearance of I: for λ > 1, min{|I ∩ Ɛ|, |I ∩ |} = 0 asymptotically almost surely, where Ɛ and are the bipartition classes of Qd, whereas for λ < 1, min{|I ∩ Ɛ|, |I ∩ |} is asymptotically almost surely exponential in d. The transition occurs in an interval whose length is of order 1/d.

A key step in the proof is an estimation of Zλ(Qd), the sum over independent sets in Qd with each set I given weight λ|I| (a.k.a. the hard-core partition function). We obtain the asymptotics of Zλ(Qd) for , and nearly matching upper and lower bounds for , extending work of Korshunov and Sapozhenko. These bounds allow us to read off some very specific information about the structure of an independent set drawn according to the hard-core distribution.

We also derive a long-range influence result. For all fixed λ > 0, if I is chosen from the independent sets of Qd according to the hard-core distribution with parameter λ, conditioned on a particular v ∈ Ɛ being in I, then the probability that another vertex w is in I is o(1) for w ∈ but Ω(1) for w ∈ Ɛ.

We prove that if K is an (infinite) stable field whose generic type has weight 1, then K is separably closed. We also obtain some partial results about stable groups and fields whose generic type has finite weight, as well as about strongly stable fields (where by definition all types have finite weight).

An (r,r+1)-factor of a graph G is a spanning subgraph H such that dH(v)∈{r,r+1} for all vertices v∈𝒱(G). If G is expressed as the union of edge-disjoint (r,r+1)-factors, then this expression is an (r,r+1)-factorization of G. Let μ(r) be the smallest integer with the property that if G is a regular loopless multigraph of degree d with d≥μ(r), then G has an (r,r+1)-factorization. It is shown that if r is even. The proof employs a novel list-coloring approach. Together with known results, this shows that μ(r)=r2+1 if r is odd and if r is even.

We prove that the eigenvalues of a certain highly non-self-adjoint operator that arises in fluid mechanics correspond, up to scaling by a positive constant, to those of a self-adjoint operator with compact resolvent; hence there are infinitely many real eigenvalues which accumulate only at ±∞. We use this result to determine the asymptotic distribution of the eigenvalues.

We prove a stability version of the Prékopa–Leindler inequality.

We apply a method of Davenport to improve several estimates for slim exceptional sets associated with the asymptotic formula in Waring’s problem. In particular, we show that the anticipated asymptotic formula in Waring’s problem for sums of seven cubes holds for all but O(N1/3+ε) of the natural numbers not exceeding N.

We consider valued fields with a value-preserving automorphism and improve on model-theoretic results by Bélair, Macintyre and Scanlon on these objects by dropping assumptions on the residue difference field. In the equicharacteristic 0 case we describe the induced structure on the value group and the residue difference field.

Let K be a convex body and let p0∈int K. Suppose that in every direction we can choose continuously a section of K which is a translated copy of the corresponding parallel section of K through p0. Our main result essentially claims that if all these pairs of sections are different almost everywhere, then K is an ellipsoid.

We introduce two graph polynomials and discuss their properties. One is a polynomial of two variables whose investigation is motivated by the performance analysis of the Bethe approximation of the Ising partition function. The other is a polynomial of one variable that is obtained by the specialization of the first one. It is shown that these polynomials satisfy deletion–contraction relations and are new examples of the V-function, which was introduced by Tutte (Proc. Cambridge Philos. Soc.43, 1947, p. 26). For these polynomials, we discuss the interpretations of special values and then obtain the bound on the number of sub-coregraphs, i.e., spanning subgraphs with no vertices of degree one. It is proved that the polynomial of one variable is equal to the monomer–dimer partition function with weights parametrized by that variable. The properties of the coefficients and the possible region of zeros are also discussed for this polynomial.

We show that on the Hilbert scheme of n points on ℂ2, the hyperkähler metric constructed by Nakajima via hyperkähler reduction is the quasi-asymptotically locally Euclidean (QALE) metric constructed by Joyce.

Since Azumaya introduced the notion of A-injectivity in 1974, several generalizations have been investigated by a number of authors. We introduce some more generalizations and discuss their connection to the previous ones.