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Let $X$ be a finite Euclidean Ramsey set. We prove that adjoining any point outside the affine hull of $X$ gives another Euclidean Ramsey set, answering a conjecture of Ivan, Leader, and Walters. We first give an elementary product proof under the additional assumptions that the orthogonal projection of the new point l...
Let $G_D(n;r)$ be the random geometric graph generated by $n$ independent uniform points on the $D$-dimensional torus, with adjacency defined by torus $L^\infty$-distance at most $r$. We study first-order zero-one and convergence laws at fixed radius and in sparse regimes. At fixed radius, we determine the asymptotics...
We prove an $\mathrm{MSO}_2$ zero-one law for a very sparse Erd\H{o}s-R\'enyi graph after pruning by component order. Let $p_n=c_n/n$, where $c_n\to0$, and delete every component of order less than $f(n)$, where $f(n)\to\infty$. If \[ f(n)\bigl(\log f(n)+\log(1/c_n)\bigr)=o(\log n), \] then the resulting graph satisfie...
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