Dingyuan Liu

On the number of regular simplices in odd dimensions

Felix Christian Clemen, Dingyuan Liu

Let $S_{d}^{k} (n)$ denote the maximum number of regular $(k - 1)$ -simplices spanned by $n$ points in $R^{d}$ . For all fixed $r \geq k \geq 3$ , the exact value of $S_{2 r}^{k} (n)$ was re…

Extremal problems for suspensions of even cycles

Dingyuan Liu

Given an integer $k \geq 2$ and a graph $F$ , the $k$ -uniform suspension $S^{k} F$ is obtained by adjoining a fixed set of $k - 2$ new vertices to every edge of $F$ . In this pape…

On the Ramsey classes of random hypergraphs

Dingyuan Liu

Let $r, s, t \geq 2$ be integers. For $r$ -graphs $G$ and $F_{1}, \dots, F_{s}$ , we write $G \to (F_{1}, \dots, F_{s})$ if every $s$ -edge-coloring of $G$ yields a monochromatic copy of $F_{i}$ in the $…

Remarks on a theorem of Erdős and Szemerédi

Dingyuan Liu

Given a graph $G$ and a real $ε > 0$ , an edge-coloring of $G$ is called $ε$ -balanced if each color appears on at least an $ε$ -fraction of the edges in $…

Ramsey problems for graphs in Euclidean spaces and Cartesian powers

Maria Axenovich, Dingyuan Liu, Arsenii Sagdeev

Given a graph $H$ , let $χ_{H} (R^{n})$ be the smallest positive integer $r$ such that there exists an $r$ -coloring of $R^{n}$ with no monochromatic unit-copy of $H$ , tha…

Turán problems for simplicial complexes

Maria Axenovich, Dániel Gerbner, Dingyuan Liu +1

An abstract simplicial complex $F$ is a non-uniform hypergraph without isolated vertices, whose edge set is closed under taking subsets. The extremal number $\mathrm{ex}(n…

The number of regular simplices in higher dimensions

Felix Christian Clemen, Adrian Dumitrescu, Dingyuan Liu

We study the extremal function $S_{d}^{k} (n)$ , defined as the maximum number of regular $(k - 1)$ -simplices spanned by $n$ points in $R^{d}$ . For any fixed $d \geq 2 k \geq 6$ , we dete…

On multiplicities of interpoint distances

Felix Christian Clemen, Adrian Dumitrescu, Dingyuan Liu

Given a set $X \subseteq R^{2}$ of $n$ points and a distance $d > 0$ , the multiplicity of $d$ is the number of times the distance $d$ appears between points in $X$ . Let $a_1(X)…

Subset selection problems in planar point sets

József Balogh, Felix Christian Clemen, Adrian Dumitrescu +1

Given a finite point set satisfying condition $A$ , the subset selection problem asks, how large of a subset satisfying condition $B$ can be extracted? In this p…

A note on the mutual-visibility coloring of hypercubes

Maria Axenovich, Dingyuan Liu

A subset $M$ of vertices in a graph $G$ is a mutual-visibility set if for any two vertices $u, v \in M$ there exists a shortest $u$ - $v$ path in $G$ that contains no elements of $M$ …

Chromatic Ramsey numbers and two-color Turán densities

Maria Axenovich, Simon Gaa, Dingyuan Liu

Given a graph $G$ , its $2$ -color Turán number $ex^{(2)} (n, G)$ is the maximum number of edges in an $n$ -vertex graph, such that the edges can be colored with two colors avo…

On the number of sets with small sumset

Dingyuan Liu, Letícia Mattos, Tibor Szabó

We investigate subsets with small sumset in arbitrary abelian groups. For an abelian group $G$ and an $n$ -element subset $Y \subseteq G$ we show that if $m ≪ s^{2} / (lo g n)^{2}$ , the…