From Tarski's plank problem to simultaneous approximation

A slab (or plank) is the part of the d-dimensional Euclidean space that lies between two parallel hyperplanes. The distance between the these hyperplanes is called the width of the slab. It is conjectured that the members of any infinite family of slabs with divergent total width can be translated so that the translates together cover the whole d-dimensional space. We prove a slightly weaker version of this conjecture, which can be regarded as a converse of Bang's theorem, also known as Tarski's plank problem. This result enables us to settle an old conjecture of Makai and Pach on simultaneous approximation of polynomials. We say that an infinite sequence S of positive numbers controls all polynomials of degree at most d if there exists a sequence of points in the plane whose x-coordinates form the sequence S, such that the graph of every polynomial of degree at most d passes within vertical distance 1 from at least one of the points. We prove that a sequence S has this property if and only if the sum of the reciprocals of the dth powers of its elements is divergent. © The Mathematical Association of America

New lower bounds for ε-nets

Following groundbreaking work by Haussler and Welzl (1987), the use of small ε-nets has become a standard technique for solving algorithmic and extremal problems in geometry and learning theory. Two significant recent developments are: (i) an upper bound on the size of the smallest ε-nets for set systems, as a function of their so-called shallow-cell complexity (Chan, Grant, Könemann, and Sharpe); and (ii) the construction of a set system whose members can be obtained by intersecting a point set in double-struck R4 by a family of half-spaces such that the size of any ε-net for them is Ω(1/ε log 1/ε) (Pach and Tardos). The present paper completes both of these avenues of research. We (i) give a lower bound, matching the result of Chan et al., and (ii) generalize the construction of Pach and Tardos to half-spaces in double-struck Rd, for any d ≥ 4, to show that the general upper bound, O(d/ε log 1/ε), of Haussler and Welzl for the size of the smallest ε-nets is tight. © Andrey Kupavskii, Nabil H. Mustafa, and János Pach

Tilings of the plane with unit area triangles of bounded diameter

There exist tilings of the plane with pairwise noncongruent triangles of equal area and bounded perimeter. Analogously, there exist tilings with triangles of equal perimeter, the areas of which are bounded from below by a positive constant. This solves a problem of Nandakumar. © 2018, Akadémiai Kiadó, Budapest, Hungary

Tilings with noncongruent triangles

We solve a problem of R. Nandakumar by proving that there is no tiling of the plane with pairwise noncongruent triangles of equal area and equal perimeter. We also show that any tiling of a convex polygon with more than three sides with finitely many triangles contains a pair of triangles that share a full side. © 2018 Elsevier Lt

On the Size of K-Cross-Free Families

Two subsets A,B of an n-element ground set X are said to be crossing, if none of the four sets A∩B, A\B, B\A and X\(A∪B) are empty. It was conjectured by Karzanov and Lomonosov forty years ago that if a family F of subsets of X does not contain k pairwise crossing elements, then |F|=O(kn). For k=2 and 3, the conjecture is true, but for larger values of k the best known upper bound, due to Lomonosov, is |F|=O(knlogn). In this paper, we improve this bound for large n by showing that |F|=Ok(nlog*n) holds, where log* denotes the iterated logarithm function. © 2018 János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Natur

Chromatic number, clique subdivisions, and the conjectures of Haj\'os and Erd\H{o}s-Fajtlowicz

For a graph $G$, let $\chi(G)$ denote its chromatic number and $\sigma(G)$ denote the order of the largest clique subdivision in $G$. Let H(n) be the maximum of $\chi(G)/\sigma(G)$ over all $n$-vertex graphs $G$. A famous conjecture of Haj\'os from 1961 states that $\sigma(G) \geq \chi(G)$ for every graph $G$. That is, $H(n) \leq 1$ for all positive integers $n$. This conjecture was disproved by Catlin in 1979. Erd\H{o}s and Fajtlowicz further showed by considering a random graph that $H(n) \geq cn^{1/2}/\log n$ for some absolute constant $c>0$. In 1981 they conjectured that this bound is tight up to a constant factor in that there is some absolute constant $C$ such that $\chi(G)/\sigma(G) \leq Cn^{1/2}/\log n$ for all $n$-vertex graphs $G$. In this paper we prove the Erd\H{o}s-Fajtlowicz conjecture. The main ingredient in our proof, which might be of independent interest, is an estimate on the order of the largest clique subdivision which one can find in every graph on $n$ vertices with independence number $\alpha$.Comment: 14 page

Acute Sets of Exponentially Optimal Size

We present a simple construction of an acute set of size (Formula presented.) in (Formula presented.) for any dimension d. That is, we explicitly give (Formula presented.) points in the d-dimensional Euclidean space with the property that any three points form an acute triangle. It is known that the maximal number of such points is less than (Formula presented.). Our result significantly improves upon a recent construction, due to Dmitriy Zakharov, with size of order (Formula presented.) where (Formula presented.) is the golden ratio. © 2018 Springer Science+Business Media, LLC, part of Springer Natur

Can Social News Websites Pay for Content and Curation? The SteemIt Cryptocurrency Model

This is an accepted manuscript of an article published by SAGE Publishing in Journal of Information Science on 15/12/2017, available online: https://doi.org/10.1177/0165551517748290 The accepted version of the publication may differ from the final published version.SteemIt is a Reddit-like social news site that pays members for posting and curating content. It uses micropayments backed by a tradeable currency, exploiting the Bitcoin cryptocurrency generation model to finance content provision in conjunction with advertising. If successful, this paradigm might change the way in which volunteer-based sites operate. This paper investigates 925,092 new members’ first posts for insights into what drives financial success in the site. Initial blog posts on average received $0.01, although the maximum accrued was$20,680.83. Longer, more sentiment-rich or more positive comments with personal information received the greatest financial reward in contrast to more informational or topical content. Thus, there is a clear financial value in starting with a friendly introduction rather than immediately attempting to provide useful content, despite the latter being the ultimate site goal. Follow-up posts also tended to be more successful when more personal, suggesting that interpersonal communication rather than quality content provision has driven the site so far. It remains to be seen whether the model of small typical rewards and the possibility that a post might generate substantially more are enough to incentivise long term participation or a greater focus on informational posts in the long term