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

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

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

On Simplices in Diameter Graphs in R-4

A graph G is a diameter graph in R-d if its vertex set is a finite subset in R-d of diameter 1 and edges join pairs of vertices a unit distance apart. It is shown that if a diameter graph G in R-4 contains the complete subgraph K on five vertices, then any triangle in G shares a vertex with K. The geometric interpretation of this statement is as follows. Given any regular unit simplex on five vertices and any regular unit triangle in R-4, then either the simplex and the triangle have a common vertex or the diameter of the union of their vertex sets is strictly greater than 1