For graphs $G$ and $H$, the Ramsey number $r(G,H)$ is the smallest positive integer $N$ such that any red/blue edge colouring of the complete graph $K_N$ contains either a red $G$ or a blue $H$. A book $B_n$ is a graph consisting of $n$ triangles all sharing a common edge.

Recently, Conlon, Fox, and Wigderson conjectured that for any $0\lt \alpha \lt 1$, the random lower bound $r(B_{\lceil \alpha n\rceil },B_n)\ge (\sqrt{\alpha }+1)^2n+o(n)$ is not tight. In other words, there exists some constant $\beta \gt (\sqrt{\alpha }+1)^2$ such that $r(B_{\lceil \alpha n\rceil },B_n)\ge \beta n$ for all sufficiently large $n$. This conjecture holds for every $\alpha \lt 1/6$ by a result of Nikiforov and Rousseau from 2005, which says that in this range $r(B_{\lceil \alpha n\rceil },B_n)=2n+3$ for all sufficiently large $n$.

We disprove the conjecture of Conlon, Fox, and Wigderson. Indeed, we show that the random lower bound is asymptotically tight for every $1/4\leq \alpha \leq 1$. Moreover, we show that for any $1/6\leq \alpha \le 1/4$ and large $n$, $r(B_{\lceil \alpha n\rceil }, B_n)\le \left (\frac 32+3\alpha \right ) n+o(n)$, where the inequality is asymptotically tight when $\alpha =1/6$ or $1/4$. We also give a lower bound of $r(B_{\lceil \alpha n\rceil }, B_n)$ for $1/6\le \alpha \lt \frac{52-16\sqrt{3}}{121}\approx 0.2007$, showing that the random lower bound is not tight, i.e., the conjecture of Conlon, Fox, and Wigderson holds in this interval.

This article attempts to develop a more systematic theoretical framework for investigating the international dissemination of devotional books in early modern times. In terms of the concept of cultural translation, the devotional genre offered fertile ground for the dynamics of selection, appropriation, decontextualization, and recontextualization. In this study, a case is made around one particular bestseller: The Practice of Piety, written by the Welsh clergyman Lewis Bayly (c.1575–1631). By studying this book's various editions and translations, we are able to consider more clearly the circumstances under which a devotional book and its textual content were governed by these dynamics. We are also able to gain greater understanding and insight into some of the actors involved: how, by whom, through which channels, and for which audiences. The primary analysis focuses on the language area of the source text: the English-speaking world. It also looks at some of the areas that, first, differ from the original context in terms of the confessional communities in which Bayly's book was translated, printed, and read; and second, for which the production, distribution, and reception of Bayly's text has been sufficiently studied, namely the Dutch- and German-language areas. The result is a premise that offers a springboard for further investigation into the dynamics at play in the international circulation of devotional books—especially in terms of text, illustration, and reading behavior.

This paper examines various ways in which apocalyptic studies can benefit from the introduction of the term and concept of gilayon, a reconstructed Hebrew counterpart of the Judeo-Greek apocalypse. The term gilayon, which combines the meanings of “revealed book” and “book of revelation,” refers to a central image of early Jewish revealed literature and could serve to define an important corpus, the boundaries of which might well overlap with (but still differ from) what is understood by the “genre apocalypse” in modern research. Moreover, this reconstructed concept uncovers additional meanings and associations, which shed light on texts known as “apocalyptic,” and has explanatory power for many phenomena associated with them. The introduction of gilayon may modify the entire paradigm of our understanding of early Jewish mysticism and help to divert the discussion of textual genres associated with it from a phenomenological to a historical route.

We consider finite simple graphs. Given a graph H and a positive integer $n,$ the Turán number of H for the order $n,$ denoted $\mathrm {ex}(n,H),$ is the maximum size of a graph of order n not containing H as a subgraph. Erdős asked: ‘For which graphs H is it true that every graph on n vertices and $\mathrm {ex}(n,H)+1$ edges contains at least two H’s? Perhaps this is always true.’ We solve this problem in the negative by proving that for every integer $k\ge 4$ there exists a graph H of order k and at least two orders n such that there exists a graph of order n and size $\mathrm {ex}(n,H)+1$ which contains exactly one copy of $H.$ Denote by $C_4$ the $4$ -cycle. We also prove that for every integer n with $6\le n\le 11$ there exists a graph of order n and size $\mathrm {ex}(n,C_4)+1$ which contains exactly one copy of $C_4,$ but, for $n=12$ or $n=13,$ the minimum number of copies of $C_4$ in a graph of order n and size $\mathrm {ex}(n,C_4)+1$ is two.