More Like this

1. Off‐Diagonal Ramsey Numbers for Slowly Growing Hypergraphs

   https://doi.org/10.1002/rsa.21284  

   Mattheus, Sam; Mubayi, Dhruv; Nie, Jiaxi; Verstraëte, Jacques (January 2025, Random Structures & Algorithms)

   ABSTRACT For ak‐uniform hypergraph and a positive integer , the Ramsey number denotes the minimum such that every ‐vertex ‐free ‐uniform hypergraph contains an independent set of vertices. A hypergraph isslowly growingif there is an ordering of its edges such that for each . We prove that if is fixed and is any non‐k‐partite slowly growing ‐uniform hypergraph, then for ,In particular, we deduce that the off‐diagonal Ramsey number is of order , where is the triple system . This is the only 3‐uniform Berge triangle for which the polynomial power of its off‐diagonal Ramsey number was not previously known. Our constructions use pseudorandom graphs and hypergraph containers. 

   more » « less
2. Tuza's conjecture for random graphs

   https://doi.org/10.1002/rsa.21057  

   Kahn, Jeff; Park, Jinyoung (November 2021, Random Structures & Algorithms)

   Abstract A celebrated conjecture of Tuza says that in any (finite) graph, the minimum size of a cover of triangles by edges is at most twice the maximum size of a set of edge‐disjoint triangles. Resolving a recent question of Bennett, Dudek, and Zerbib, we show that this is true for random graphs; more precisely:urn:x-wiley:rsa:media:rsa21057:rsa21057-math-0001 

   more » « less
3. The local solubility for homogeneous polynomials with random coefficients over thin sets

   https://doi.org/10.1112/mtk.12282  

   Lee, Heejong; Lee, Seungsu; Yeon, Kiseok (October 2024, Mathematika)

   Abstract Let and be natural numbers greater or equal to 2. Let be a homogeneous polynomial in variables of degree with integer coefficients , where denotes the inner product, and denotes the Veronese embedding with . Consider a variety in , defined by . In this paper, we examine a set of integer vectors , defined bywhere is a nonsingular form in variables of degree with for some constant depending at most on and . Suppose has a nontrivial integer solution. We confirm that the proportion of integer vectors in , whose associated equation  is everywhere locally soluble, converges to a constant as . Moreover, for each place of , if there exists a nonzero such that and the variety in admits a smooth ‐point, the constant is positive. 

   more » « less
4. Stability of the catenoid for the hyperbolic vanishing mean curvature equation outside symmetry

   https://doi.org/10.1007/s00222-025-01330-3  

   Lührmann, Jonas; Oh, Sung-Jin; Shahshahani, Sohrab (March 2025, Inventiones mathematicae)

   Abstract We study the problem of stability of the catenoid, which is an asymptotically flat rotationally symmetric minimal surface in Euclidean space, viewed as a stationary solution to the hyperbolic vanishing mean curvature equation in Minkowski space. The latter is a quasilinear wave equation that constitutes the hyperbolic counterpart of the minimal surface equation in Euclidean space. Our main result is the nonlinear asymptotic stability, modulo suitable translation and boost (i.e., modulation), of the$$n$$ $n$-dimensional catenoid with respect to a codimension one set of initial data perturbations without any symmetry assumptions, for$$n \geq 5$$ $n\ge 5$. The modulation and the codimension one restriction on the data are necessary and optimal in view of the kernel and the unique simple eigenvalue, respectively, of the stability operator of the catenoid. In a broader context, this paper fits in the long tradition of studies of soliton stability problems. From this viewpoint, our aim here is to tackle some new issues that arise due to the quasilinear nature of the underlying hyperbolic equation. Ideas introduced in this paper include a new profile construction and modulation analysis to track the evolution of the translation and boost parameters of the stationary solution, a new scheme for proving integrated local energy decay for the perturbation in the quasilinear and modulation-theoretic context, and an adaptation of the vectorfield method in the presence of dynamic translations and boosts of the stationary solution. 

   more » « less
5. On a problem of M. Talagrand

   https://doi.org/10.1002/rsa.21077  

   Frankston, Keith; Kahn, Jeff; Park, Jinyoung (December 2022, Random Structures & Algorithms)

   Abstract We address a special case of a conjecture of M. Talagrand relating two notions of “threshold” for an increasing family of subsets of a finite setV. The full conjecture implies equivalence of the “Fractional Expectation‐Threshold Conjecture,” due to Talagrand and recently proved by the authors and B. Narayanan, and the (stronger) “Expectation‐Threshold Conjecture” of the second author and G. Kalai. The conjecture under discussion here says there is a fixedLsuch that if, for a given , admits withand(a.k.a.  isweakly p‐small), then admits such a taking values in ( is‐small). Talagrand showed this when is supported on singletons and suggested, as a more challenging test case, proving it when is supported on pairs. The present work provides such a proof. 

   more » « less