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The canonical van der Waerden theorem asserts that, for sufficiently large n, every colouring of [n] contains either a monochromatic or a rainbow arithmetic progression of length k (k-AP, for short). In this paper, we determine the threshold at which the binomial random subset [n]_p almost surely inherits this canonical Ramsey type property. As an application, we show the existence of sets A⊆ [n] such that the k-APs in A define a k-uniform hypergraph of arbitrarily high girth and yet any colouring of A induces a monochromatic or rainbow k-AP.
The real log canonical threshold (RLCT) is a central invariant in birational geometry and singularity theory, measuring the complexity of a singularity through discrepancy and valuation data on a log resolution. Beyond this algebro-geometric definition, it also admits a metric interpretation, reflecting how neighbourhoods of the singular locus degenerate at small scales. In this work, we investigate these degenerations via sublevel sets associated with an analytic ideal. We show that the asymptotic behaviour of their volume determines the visible intrinsic divisorial spectrum (i.e. the set of actual poles of the local zeta function), a finite set contained in the resolution-dependent set of multiplicity ratios of any log resolution. Conversely, this intrinsic spectrum, together with its multiplicities and coefficients, can be recovered from the volume function through a finite reconstruction procedure. We also describe intrinsic interpretations in terms of arc spaces: the divisorial exponents appear both as ratios of vanishing orders along generic arcs and as asymptotic codimension growth rates of divisorial cylinders. Taken together, these results show that certain divisorial invariants admit a metric realisation through the asymptotic behaviour of sublevel-set volumes, and that the birational structure of an analytic singularity can be reconstructed from the geometry of its infinitesimal neighbourhoods.
A Gallai k-colouring of a graph G is a colouring of E(G) with k colours that induces no rainbow triangles, that is, a triangle with edges of 3 different colours. We give a first step towards estimating the number of Gallai colourings of the Erdős-Rényi random graph, by proving that for every δ> 0 there are c and C such that with high probability the number of Gallai 3-colourings of G(n,p) is at least 3^(1-δ)n2p for p ≤ cn^-1/2, and at most 2^(1+δ)n2p for p ≥ Cn^-1/2.
We construct one-parameter deformations of the Euclidean sphere S-n inside Rn+1 that admit a Zoll family of codimension one embedded minimal spheres, in all dimensions n >= 2. For n = 2, this means that all geodesics are closed, embedded, and of the same period. The method of construction is equivariant with respect to the natural actions of the orthogonal group. In particular, we show that the original Zoll spheres of revolution in R-3 have counterparts in the context of minimal surface theory, in all dimensions. We also describe the first examples of metrics on the real projective spaces RPn, in all dimensions n >= 3, that admit a Zoll family of embedded minimal projective hyperplanes, and which are not isometric to metrics with minimal linear projective hyperplanes. The new constructions are underpinned by equivariant versions of Nash-Moser-Hamilton implicit function theorem, and yield new information even in dimension n = 2. As an application, we also show that every finite group of the orthogonal group O(3) that does not contain-Id is the isometry group of some (classical) Zoll metric on S-2. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We establich quantitative stability estimates for the Trudinger-Moser inequality on smooth, bounded domains in the Euclidean plane. More specifically, we prove that the deficit in the Trudinger-Moser inequality quadratically controls the distance to the set of optimizers if either (i) the exponential rate of growth is sufficiently small or (ii) the domain is a round disk. The latter estimate remains valid even in the critical case. Both proofs rely on a new spectral gap that we prove, which may be of independent interest. Additionally we show that the same stability estimate holds in the nondegenerate case, and that this occurs generically.
