Add to ORKGLet $\De$ be a non-degenerate simplex on $k$ vertices. We prove that there exists a threshold $s_k<k$ such that any set $A\subs \R^k$ of Hausdorff dimension $dim\,A\geq s_k$ necessarily contains a similar copy of the simplex $\De$.Comment: arXiv admin note: text overlap with arXiv:2006.1572
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Let $d$ be an integer, and let $E$ be a nonempty closed subset of $\mathbb{R}^n$. Assume that $E$ is...
A $d$-simplex is defined to be a collection $A_1,\dots,A_{d+1}$ of subsets of size $k$ of $[n]$ such...
In this paper, we first show that for all four non-negative real numbers, there exists a Cantor ultr...
In this paper we show that if a compact set $E \subset \mathbb{R}^d$, $d \geq 3$, has Hausdorff dime...
We recursively extend the Lov\'asz theta number to geometric hypergraphs on the unit sphere and on E...
http://arxiv.org/PS_cache/math/pdf/0703/0703504v2.pdfWe prove that a sufficiently large subset of th...
A theorem of Steinhaus states that if $E\subset \mathbb R^d$ has positive Lebesgue measure, then the...
Abstract. In a recent paper, Chan, Laba, and Pramanik investigated geometric configurations inside ...
Let $A$ and $B$ be Borel subsets of the Euclidean $n$-space with $\dim A + \dim B > n$. This is a su...
AbstractIn answer to a question of Erdős, it is shown that, given any triangle, there exists a subse...
We prove that there exists a constant $\varepsilon > 0$ with the following property: if $K \subset \...
Funding: Alexia Yavicoli was financially supported by the Swiss National Science Foundation, grant n...
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Our first main result is a uniform bound, in every dimension $k \in \mathbb N$, on the topological T...
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