solidot新版网站常见问题,请点击这里查看。
消息
本文已被查看114次
Distinct Volume Subsets via Indiscernibles. (arXiv:1807.06654v1 [math.LO])
来源于:arXiv
Erd\"{o}s proved that for every infinite $X \subseteq \mathbb{R}^d$ there is
$Y \subseteq X$ with $|Y|=|X|$, such that all pairs of points from $Y$ have
distinct distances, and he gave partial results for general $a$-ary volume. In
this paper, we search for the strongest possible canonization results for
$a$-ary volume, making use of general model-theoretic machinery. The main
difficulty is for singular cardinals; to handle this case we prove the
following. Suppose $T$ is a stable theory, $\Delta$ is a finite set of formulas
of $T$, $M \models T$, and $X$ is an infinite subset of $M$. Then there is $Y
\subseteq X$ with $|Y| = |X|$ and an equivalence relation $E$ on $Y$ with
infinitely many classes, each class infinite, such that $Y$ is $(\Delta,
E)$-indiscernible. We also consider the definable version of these problems,
for example we assume $X \subseteq \mathbb{R}^d$ is perfect (in the topological
sense) and we find some perfect $Y \subseteq X$ with all distances distinct.
Finally we 查看全文>>