The pentagram map was introduced by Schwartz in 1992 for convex planar polygons. Recently, Ovsienko, Schwartz, and Tabachnikov proved Liouville integrability of the pentagram map for generic monodromies by providing a Poisson structure and the sufficient number of integrals in involution on the space of twisted polygons.

In this paper we prove algebraic-geometric integrability for any monodromy, that is, for both twisted and closed polygons. For that purpose we show that the pentagram map can be written as a discrete zero-curvature equation with a spectral parameter, and we study the corresponding spectral curve and the dynamics on its Jacobian. We also prove that on the symplectic leaves Poisson brackets discovered for twisted polygons coincide with the symplectic structure obtained from Krichever–Phong’s universal formula.

This is the first of a series of articles on complete Calabi–Yau manifolds asymptotic to Riemannian cones at infinity. We begin by proving general existence and uniqueness results. The uniqueness part relaxes the decay condition $O\left(r^{-n-\epsilon }\right)$ needed in earlier work to $O\left(r^{-\epsilon }\right)$, relying on some new ideas about harmonic functions. We then look at two classes of examples: crepant resolutions of cones (this includes a new class of Ricci-flat small resolutions associated with flag varieties) and affine deformations of cones. One focus here is the question of the precise rate of decay of the metric to its tangent cone. We prove that the optimal rate for the Stenzel metric on $T^{\ast }{S}^n$ is $-2(n/(n-1\left)\right)$.

Ramsey’s theorem, in the version of Erdős and Szekeres, states that every $2$-coloring of the edges of the complete graph on $\{1,2,\dots ,n\}$ contains a monochromatic clique of order $(1/2)logn$. In this article, we consider two well-studied extensions of Ramsey’s theorem. Improving a result of Rödl, we show that there is a constant $c>0$ such that every $2$-coloring of the edges of the complete graph on $\{2,3,\dots ,n\}$ contains a monochromatic clique $S$ for which the sum of $1/logi$ over all vertices $i\in S$ is at least $clogloglogn$. This is tight up to the constant factor $c$ and answers a question of Erdős from 1981. Motivated by a problem in model theory, Väänänen asked whether for every $k$ there is an $n$ such that the following holds: for every permutation $\pi$ of $1,\dots ,k-1$, every $2$-coloring of the edges of the complete graph on $\{1,2,\dots ,n\}$ contains a monochromatic clique $a_1<\cdots <a_k$ with

$$a_{\pi \left(1\right)+1}-a_{\pi \left(1\right)}>a_{\pi \left(2\right)+1}-a_{\pi \left(2\right)}>\cdots >a_{\pi (k-1)+1}-a_{\pi (k-1)}.$$

That is, not only do we want a monochromatic clique, but the differences between consecutive vertices must satisfy a prescribed order. Alon and, independently, Erdős, Hajnal, and Pach answered this question affirmatively. Alon further conjectured that the true growth rate should be exponential in $k$. We make progress towards this conjecture, obtaining an upper bound on $n$ which is exponential in a power of $k$. This improves a result of Shelah, who showed that $n$ is at most double-exponential in $k$.

A mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies. We compute a mapping class group of a hyperkähler manifold $M$, showing that it is commensurable to an arithmetic lattice in $SO(3,b_2-3)$. A Teichmüller space of $M$ is a space of complex structures on $M$ up to isotopies. We define a birational Teichmüller space by identifying certain points corresponding to bimeromorphically equivalent manifolds. We show that the period map gives the isomorphism between connected components of the birational Teichmüller space and the corresponding period space $SO(b_2-3,3)/SO\left(2\right)\times SO(b_2-3,1)$. We use this result to obtain a Torelli theorem identifying each connected component of the birational moduli space with a quotient of a period space by an arithmetic group. When $M$ is a Hilbert scheme of $n$ points on a K3 surface, with $n-1$ a prime power, our Torelli theorem implies the usual Hodge-theoretic birational Torelli theorem (for other examples of hyperkähler manifolds, the Hodge-theoretic Torelli theorem is known to be false).

Our earlier published article (vol. 159, no. 3, pp. 385–415) had a gap in the proof of Proposition 6.11. The problem was that, in order to deduce (17) from (16), the old proof used identity $2E_{j,m+1}^n$ rather than identity $2E_{j,m}^n$, as originally stated. In this erratum, we correct the error by giving a direct proof of Theorem 6.9(2) (using the numbering of the previous paper). Starting with Theorem 6.9 of our previous paper, the second half of Section 6 should be replaced by the text which follows.