@article{14660, abstract = {The classical Steinitz theorem states that if the origin belongs to the interior of the convex hull of a set π‘†βŠ‚β„π‘‘, then there are at most 2𝑑 points of 𝑆 whose convex hull contains the origin in the interior. BΓ‘rΓ‘ny, Katchalski,and Pach proved the following quantitative version of Steinitz’s theorem. Let 𝑄 be a convex polytope in ℝ𝑑 containing the standard Euclidean unit ball 𝐁𝑑. Then there exist at most 2𝑑 vertices of 𝑄 whose convex hull 𝑄′ satisfies π‘Ÿππ‘‘βŠ‚π‘„β€² with π‘Ÿβ©Ύπ‘‘βˆ’2𝑑. They conjectured that π‘Ÿβ©Ύπ‘π‘‘βˆ’1βˆ•2 holds with a universal constant 𝑐>0. We prove π‘Ÿβ©Ύ15𝑑2, the first polynomial lower bound on π‘Ÿ. Furthermore, we show that π‘Ÿ is not greater than 2/βˆšπ‘‘.}, author = {Ivanov, Grigory and NaszΓ³di, MΓ‘rton}, issn = {1469-2120}, journal = {Bulletin of the London Mathematical Society}, publisher = {London Mathematical Society}, title = {{Quantitative Steinitz theorem: A polynomial bound}}, doi = {10.1112/blms.12965}, year = {2023}, } @article{11186, abstract = {In this note, we study large deviations of the number 𝐍 of intercalates ( 2Γ—2 combinatorial subsquares which are themselves Latin squares) in a random 𝑛×𝑛 Latin square. In particular, for constant 𝛿>0 we prove that exp(βˆ’π‘‚(𝑛2log𝑛))β©½Pr(𝐍⩽(1βˆ’π›Ώ)𝑛2/4)β©½exp(βˆ’Ξ©(𝑛2)) and exp(βˆ’π‘‚(𝑛4/3(log𝑛)))β©½Pr(𝐍⩾(1+𝛿)𝑛2/4)β©½exp(βˆ’Ξ©(𝑛4/3(log𝑛)2/3)) . As a consequence, we deduce that a typical order- 𝑛 Latin square has (1+π‘œ(1))𝑛2/4 intercalates, matching a lower bound due to Kwan and Sudakov and resolving an old conjecture of McKay and Wanless.}, author = {Kwan, Matthew Alan and Sah, Ashwin and Sawhney, Mehtaab}, issn = {1469-2120}, journal = {Bulletin of the London Mathematical Society}, number = {4}, pages = {1420--1438}, publisher = {Wiley}, title = {{Large deviations in random latin squares}}, doi = {10.1112/blms.12638}, volume = {54}, year = {2022}, } @article{9572, abstract = {We prove that every n-vertex tournament G has an acyclic subgraph with chromatic number at least n5/9βˆ’o(1), while there exists an n-vertex tournament G whose every acyclic subgraph has chromatic number at most n3/4+o(1). This establishes in a strong form a conjecture of Nassar and Yuster and improves on another result of theirs. Our proof combines probabilistic and spectral techniques together with some additional ideas. In particular, we prove a lemma showing that every tournament with many transitive subtournaments has a large subtournament that is almost transitive. This may be of independent interest.}, author = {Fox, Jacob and Kwan, Matthew Alan and Sudakov, Benny}, issn = {1469-2120}, journal = {Bulletin of the London Mathematical Society}, number = {2}, pages = {619--630}, publisher = {Wiley}, title = {{Acyclic subgraphs of tournaments with high chromatic number}}, doi = {10.1112/blms.12446}, volume = {53}, year = {2021}, } @article{6965, abstract = {The central object of investigation of this paper is the Hirzebruch class, a deformation of the Todd class, given by Hirzebruch (for smooth varieties). The generalization for singular varieties is due to Brasselet–SchΓΌrmann–Yokura. Following the work of Weber, we investigate its equivariant version for (possibly singular) toric varieties. The local decomposition of the Hirzebruch class to the fixed points of the torus action and a formula for the local class in terms of the defining fan are recalled. After this review part, we prove the positivity of local Hirzebruch classes for all toric varieties, thus proving false the alleged counterexample given by Weber.}, author = {Rychlewicz, Kamil P}, issn = {1469-2120}, journal = {Bulletin of the London Mathematical Society}, number = {2}, pages = {560--574}, publisher = {Wiley}, title = {{The positivity of local equivariant Hirzebruch class for toric varieties}}, doi = {10.1112/blms.12442}, volume = {53}, year = {2021}, } @article{9573, abstract = {It is a classical fact that for any Ξ΅>0, a random permutation of length n=(1+Ξ΅)k2/4 typically contains a monotone subsequence of length k. As a far-reaching generalization, Alon conjectured that a random permutation of this same length n is typically k-universal, meaning that it simultaneously contains every pattern of length k. He also made the simple observation that for n=O(k2logk), a random length-n permutation is typically k-universal. We make the first significant progress towards Alon's conjecture by showing that n=2000k2loglogk suffices.}, author = {He, Xiaoyu and Kwan, Matthew Alan}, issn = {1469-2120}, journal = {Bulletin of the London Mathematical Society}, number = {3}, pages = {515--529}, publisher = {Wiley}, title = {{Universality of random permutations}}, doi = {10.1112/blms.12345}, volume = {52}, year = {2020}, }