Related papers: Zarankiewicz bounds from distal regularity lemma

Let $S$ be a set of $n\times n$ matrices over a field $\mathbb{F}$. We show that the $\mathbb{F}$-linear span of the words in $S$ of length at most $$2n\log_2n+4n$$ is the full $\mathbb{F}$-algebra generated by $S$. This improves on the…

Some results of B. Pasynkov and H. Torunczyk on finite-dimensional maps are improved. A generalization of a Dranishnikov-Uspenskij theorem about extensional dimension is also obtained.

There are a couple of proofs by now for the famous Cwikel--Lieb--Rozenblum (CLR) bound, which is a semiclassical bound on the number of bound states for a Schr\"odinger operator, proven in the 1970s. Of the rather distinct proofs by Cwikel,…

The best degree-based upper bound for the spectral radius is due to Liu and Weng. This paper begins by demonstrating that a (forgotten) upper bound for the spectral radius dating from 1983 is equivalent to their much more recent bound. This…

Gallagher's theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto,…

We study large random partitions boxed into a rectangle and coming from skew Howe duality, or alternatively from dual Schur measures. As the sides of the rectangle go to infinity, we obtain: 1) limit shape results for the profiles…

Fox, Gromov, Lafforgue, Naor, and Pach proved a regularity lemma for semi-algebraic $k$-uniform hypergraphs of bounded complexity, showing that for each $\epsilon>0$ the vertex set can be equitably partitioned into a bounded number of parts…

We prove boundary H\"older and Lipschitz regularity for a class of degenerate elliptic, second order, inhomogeneous equations in non-divergence form structured on the left-invariant vector fields of the Heisenberg group. Our focus is on the…

We consider a family of self-adjoint Ornstein--Uhlenbeck operators $L_{\alpha} $ in an infinite dimensional Hilbert space H having the same gaussian invariant measure $\mu$ for all $\alpha \in [0,1]$. We study the Dirichlet problem for the…

Let $\lambda_k$ denote the $k$-th successive minimum of a lattice $L$. We study properties of the lengths of certain bases of $L$. If $v_1, \dots v_n$ is a basis which is reduced in the sense of Minkowski we show that $\lvert v_k \rvert^2…

In 1994, Erd\H{o}s, Gy\'{a}rf\'{a}s and {\L}uczak posed the following problem: given disjoint vertex sets $V_1,\dots,V_n$ of size~$k$, with exactly one edge between any pair $V_i,V_j$, how large can $n$ be such that there will always be an…

Twin-width is a structural width parameter introduced by Bonnet, Kim, Thomass\'e and Watrigant [FOCS 2020], and has interesting applications in the areas of logic on graphs and in parameterized algorithmics. Very briefly, the essence of…

For a smooth $k$-dimensional submanifold $\Sigma$ of a $d$-dimensional compact Riemannian manifold $M$, we extend the $L^p(\Sigma)$ restriction bounds of Burq-G\'erard-Tzvetkov -- originally proved for individual Laplace--Beltrami…

It is well-known that if $(A,B)$ is an $\tfrac{\varepsilon}{2}$-regular pair (in the sense of Szemer\'edi) then there exist sets $A'\subset A$ and $B'\subset B'$ with $|A'|\leq \varepsilon|A|$ and $|B'|\leq \varepsilon|B|$ so that the…

Let $G$ be a connected nonregular graphs of order $n$ with maximum degree $\Delta$ that attains the maximum spectral radius. Liu and Li (2008) proposed a conjecture stating that $G$ has a degree sequence $(\Delta,\ldots,\Delta,\delta)$ with…

Szpilrajn's Lemma entails that each partial order extends to a linear order. Dushnik and Miller use Szpilrajn's Lemma to show that each partial order has a relizer. Since then, many authors utilize Szpilrajn's Theorem and the Well-ordering…

The problem of constructing or characterizing strongly regular Cayley graphs (or equivalently, regular partial difference sets) has garnered significant attention over the past half-century. In 2003, Miklavi\v{c} and Poto\v{c}nik [European…

We prove explicit bounds on the radius of a ball centered at the origin which is guaranteed to contain all bounded connected components of a semi-algebraic set $S \subset \mathbbm{R}^k$ defined by a quantifier-free formula involving $s$…

A graph $G$ is universal for a class of graphs $\mathcal{C}$, if, up to isomorphism, $G$ contains every graph in $\mathcal{C}$ as a subgraph. In 1978, Chung and Graham asked for the minimal number $s(n)$ of edges in a graph with $n$…

We show new upper and lower bounds for the effective differential Nullstellensatz for differential fields of characteristic zero with several commuting derivations. Seidenberg was the first to address this problem in 1956, without giving a…