arXiv subjects
Publications and source records attributed to Zion Hefty.
We give a nibble-free construction proving $R(3,k)\ge(1/200+o(1))k^2/\log k$. We also include Shearer's proof bounding the independence number of a triangle-free graph, which implies $R(3,k)\le (1+o(1))(k^2/\log k)$.
We present a flexible random construction which, for certain graphs $H$, is able to produce $H$-free graphs with edge density strictly larger than that of the $H$-free process, while simultaneously preserving pseudorandom properties and allowing a much easier analysis. As our main application, we use this construction to show that the off-diagonal Ramsey numbers satisfy $R(3,k)\ge \left(\frac12+o(1)\right)\frac{k^2}{\log{k}}$, improving the previously best bound $R(3,k)\ge \left(\frac13+o(1)\right)\frac{k^2}{\log{k}}$. While the best known upper bound is $R(3,k)\le \left(1+o(1)\right)\frac{k^2}{\log{k}}$, the constant of $\frac12$ has been conjectured to be asymptotically tight by multiple groups.
Given a number field $K$ that is a subfield of the real numbers, we generalize the notion of the classical Frobenius problem to the ring of integers $\mathfrak{O}_K$ of $K$ by describing certain Frobenius semigroups, $\mathrm{Frob}(\alpha_1,\dots,\alpha_n)$, for appropriate elements $\alpha_1,\dots,\alpha_n\in\mathfrak{O}_K$. We construct a partial ordering on $\mathrm{Frob}(\alpha_1,\dots,\alpha_n)$, and show that this set is completely described by the maximal elements with respect to this ordering. We also show that $\mathrm{Frob}(\alpha_1,\dots,\alpha_n)$ will always have finitely many such maximal elements, but in general, the number of maximal elements can grow without bound as $n$ is fixed and $\alpha_1,\dots,\alpha_n\in\mathfrak{O}_K$ vary. Explicit examples of the Frobenius semigroups are also calculated for certain cases in real quadratic number fields.