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Dingyuan Liu

Publications and source records attributed to Dingyuan Liu.

15 recordsLinked to original sources
arXiv2026

Subset selection problems in planar point sets

Given a finite point set satisfying condition $\mathcal{A}$, the subset selection problem asks, how large of a subset satisfying condition $\mathcal{B}$ can be extracted? In this paper, we make progress on three instances of subset selection problems in planar point sets. Let $n,s\in\mathbb{N}$ with $n\geq s$, and let $P\subseteq\mathbb{R}^2$ be a set of $n$ points, where at most $s$ points lie on the same line. Firstly, we select a general position subset of $P$. This problem was proposed by Erdős under the regime when $s$ is a constant. For $s$ being non-constant, we give new lower and upper bounds on the maximum size of such a subset. In particular, we show that in the worst case such a set can have size at most $O(n^{5/6+o(1)}/\sqrt{s})$ when $3\leq s\leq n^{1/3}$ and $O(n/s)$ when $n^{1/3}\leq s\leq n$. Secondly, we select a monotone general position subset of $P$, that is, a subset in general position where the points are ordered from left to right and their $y$-coordinates are either non-decreasing or non-increasing. We present bounds on the maximum size of such a subset. In particular, when $s=Ω(\sqrt{n})$, our upper and lower bounds differ at most by a logarithmic factor. Lastly, we select a subset of $P$ with pairwise distinct slopes. This problem was initially studied by Erdős, Graham, Ruzsa, and Taylor on the grid. We show that for $s=O(\sqrt{n})$ such a subset of size $Ω((n/\log{s})^{1/3})$ can always be found in $P$. When $s=Θ(\sqrt{n})$, this matches a lower bound given by Zhang on the grid. As for the upper bound, we show that in the worst case such a subset has size at most $O(\sqrt{n})$ for $2\leq s\leq n^{3/8}$ and $O((n/s)^{4/5})$ for $n^{3/8}\leq s=O(\sqrt{n})$. The proofs use a wide range of tools such as incidence geometry, probabilistic methods, the hypergraph container method, and additive combinatorics.

math.CO↗
arXiv2026

Extremal problems for suspensions of even cycles

Given an integer $k\geq2$ and a graph $F$, the $k$-uniform suspension $\mathcal{S}^kF$ is obtained by adjoining a fixed set of $k-2$ new vertices to every edge of $F$. In this paper, we study two extremal problems for suspensions of even cycles. Write $K^k_t$ for the $k$-uniform clique of order $t$. Let $\mathrm{ex}(n,\mathcal{S}^kC_{2\ell})$ and $\mathrm{ex}(n,K^k_t,\mathcal{S}^kC_{2\ell})$ denote the maximum numbers of edges and copies of $K^k_t$, respectively, in an $\mathcal{S}^kC_{2\ell}$-free $k$-uniform hypergraph on $n$ vertices. We prove that, for every $k\geq2$ and infinitely many $n$, \[\mathrm{ex}(n,K^{k}_{k+1},\mathcal{S}^kC_4)=\frac{n^{k-1/2}}{(k+1)!}+O(n^{k-1}).\] This extends a folklore result for $k=2$ and, as an immediate consequence, yields the asymptotics of $\mathrm{ex}(n,\mathcal{S}^kC_4)$ for infinitely many $n$, previously established by Mubayi (for all $n$). Furthermore, for every $k\geq2$ and $\ell\in\{3,5\}$, we determine the order of magnitude \[\mathrm{ex}(n,\mathcal{S}^kC_{2\ell})=Θ(n^{k-1+1/\ell}).\] This generalizes both the classical graph case $k=2$ and a previous result of Mukherjee for $k=\ell=3$. The principal difficulty in both problems lies in constructing the lower bounds. Our construction for $\mathrm{ex}(n,K^{k}_{k+1},\mathcal{S}^kC_4)$ incorporates a novel block-packing structure, which yields substantially more copies of $K^k_{k+1}$ than previously known constructions. For $\mathrm{ex}(n,\mathcal{S}^kC_{2\ell})$ with $\ell\in\{3,5\}$, we establish a natural $k$-uniform version of Wenger graphs, addressing the subtleties involved in lifting extremal graph constructions to suspensions. We also give applications of our results to Turán problems for simplicial complexes.

math.CO↗
arXiv2026

Turán problems for simplicial complexes

An abstract simplicial complex $\mathbf{F}$ is a non-uniform hypergraph without isolated vertices, whose edge set is closed under taking subsets. The extremal number $\mathrm{ex}(n,\mathbf{F})$ is defined as the maximum number of edges in an $n$-vertex $\mathbf{F}$-free simplicial complex. Although Turán-type problems for simplicial complexes have long appeared in extremal set theory, a systematic study of $\mathrm{ex}(n,\mathbf{F})$ was initiated only recently by Conlon, Piga, and Schülke. In contrast to uniform hypergraphs, even the order of magnitude of $\mathrm{ex}(n,\mathbf{F})$ remains unknown for most simplicial complexes. In this paper, we present a general framework for estimating $\mathrm{ex}(n,\mathbf{F})$ via generalised Turán numbers of associated hypergraphs. Using this approach, we determine the asymptotic behaviour, and in some cases the exact value, of $\mathrm{ex}(n,\mathbf{F})$ for broad classes of simplicial complexes, extending a result of Conlon, Piga, and Schülke. We also exhibit simplicial complexes whose extremal numbers are not governed by the corresponding generalised Turán numbers, and determine their extremal numbers asymptotically. In addition, we study how the extremal number changes when a new edge is added to the forbidden simplicial complex, and obtain a tight bound for this behaviour.

math.CO↗
arXiv2026

The number of regular simplices in higher dimensions

We study the extremal function $S^k_d(n)$, defined as the maximum number of regular $(k-1)$-simplices spanned by $n$ points in $\mathbb{R}^d$. For any fixed $d\geq2k\geq6$, we determine the asymptotic behavior of $S^k_d(n)$ up to a lower-order term. In particular, when $k=3$, we determine the exact value of $S^3_d(n)$, for all even dimensions $d\geq6$ and sufficiently large $n$. This resolves a conjecture of Erdős in a stronger form. The proof leverages techniques from hypergraph Turán theory and linear algebra.

math.CO↗
arXiv2026

Chromatic Ramsey numbers and two-color Turán densities

Given a graph $G$, its $2$-color Turán number $\mathrm{ex}^{(2)}(n,G)$ is the maximum number of edges in an $n$-vertex graph, such that the edges can be colored with two colors avoiding a monochromatic copy of $G$. Let $π^{(2)}(G)=\lim_{n\to\infty}\mathrm{ex}^{(2)}(n,G)/\binom{n}{2}$ be the $2$-color Turán density of $G$. What real numbers in the interval $(0,1)$ are realized as the $2$-color Turán density of some graph? It is known that $π^{(2)}(G)=1-(R_χ(G)-1)^{-1}$, where $R_χ(G)$ is the chromatic Ramsey number of $G$. Burr, Erdős, and Lovász showed that $(k-1)^2+1\leq{R_χ(G)}\leq{R(k)}$, for any $k$-chromatic graph $G$, where $R(k)$ is the classical Ramsey number. However, it is an open problem to determine how many distinct values between $(k-1)^{2}+1$ and $R(k)$ can be realized as $R_χ(G)$ of some $k$-chromatic graph $G$ for general $k$. In this paper, among others, we prove that there are $Ω(k)$ different values of $R_χ(G)$ among $k$-chromatic graphs $G$. This sheds more light onto the possible $2$-color Turán densities of graphs.

math.CO↗
arXiv2026

On the Ramsey classes of random hypergraphs

Let $r,s,t\geq2$ 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 $i$-th color for some $1\leq i\leq s$. Let $\mathcal{R}(F_1,\dots,F_s)$ denote the family of all $r$-graphs $G$ with $G\to(F_1,\dots,F_s)$. When $F_1=\dots=F_s=F$, we write $\mathcal{R}(F;s)=\mathcal{R}(F_1,\dots,F_s)$. In this paper, we investigate when $\mathcal{R}(H;s)\subseteq\mathcal{R}(Q_1,\dots,Q_t)$ holds, where $H=H^{(r)}(n,p)$ is a random $r$-graph and $Q_1,\dots,Q_t$ are fixed $r$-graphs. Our main result determines the threshold for a large class of such $Q_1,\dots,Q_t$, including complete $r$-graphs. The key ingredient in our proof is a generalization of a result of Graham, Łuczak, Rödl, and Ruciński, which provides a necessary and sufficient condition for $\mathcal{R}(F_1,\dots,F_s)\subseteq\mathcal{R}(Q_1,\dots,Q_t)$, where $Q_1,\dots,Q_t$ are highly connected. As a byproduct, we characterize when two tuples of highly connected $r$-graphs are Ramsey equivalent.

math.CO↗
arXiv2026

Could Large Language Models work as Post-hoc Explainability Tools in Credit Risk Models?

Large language models (LLMs) have shown promise in translating model-based explanations into human-readable narratives. This study evaluates whether LLMs can serve as post-hoc explainability interfaces for credit risk models, focusing on their ability to preserve feature-importance rankings and generate autonomous explanations. Using a LendingClub dataset, we compare LLM outputs with SHAP and coefficient-based attributions on three major LLMs, including GPT-4-turbo, Claude-Sonnet-4.5, and Gemini-2.5-Flash. Results indicate that LLMs reliably reproduce reference rankings under controlled prompts but show limited alignment when generating explanations autonomously. These findings suggest that LLMs are best deployed as narrative interfaces rather than substitutes for formal attribution methods in credit risk governance.

q-fin.RM↗
arXiv2026

Visibility in hypercubes

A subset $M$ of vertices in a graph $G$ is a mutual-visibility set if any two vertices $u$ and $v$ in $M$ ``see'' each other in $G$, that is, there exists a shortest $u,v$-path in $G$ that contains no elements of $M$ as internal vertices. The mutual-visibility number $μ(G)$ of a graph $G$ is the largest size of a mutual-visibility set in $G$. Let $n\in\mathbb{N}$ and $Q_{n}$ be an $n$-dimensional hypercube. Cicerone, Di Fonso, Di Stefano, Navarra, and Piselli showed that $2^{n}/\sqrt{n}\leqμ(Q_{n})\leq2^{n-1}$. In this paper, we prove that $μ(Q_{n})>0.186\cdot2^n$ and thus establish that $μ(Q_{n})=Θ(2^{n})$. We also consider the chromatic mutual-visibility number, $χ_μ(G)$, defined as the smallest number of colors used on vertices of $G$, such that every color class is a mutual-visibility set in $G$. Klavžar, Kuziak, Valenzuela-Tripodoro, and Yero asked whether $χ_μ(Q_{n})=O(1)$. We answer their question in the negative, namely, we show that $χ_μ(Q_{n})$ is a growing function of $n$. Moreover, we show that $χ_μ(Q_{n})=O(\log\log{n})$. Finally, we study the so-called total mutual-visibility number of graphs and give asymptotically tight bounds on this parameter for hypercubes.

math.CO↗
arXiv2026

A note on the mutual-visibility coloring of hypercubes

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 $M$ as internal vertices. Let $χ_μ(G)$ be the least number of colors needed to color the vertices of $G$, so that each color class is a mutual-visibility set. Let $n\in\mathbb{N}$ and $Q_{n}$ be an $n$-dimensional hypercube. It was proved by the authors that the maximum size of a mutual-visibility set in $Q_{n}$ is at least $Ω(2^{n})$. Klavžar, Kuziak, Valenzuela-Tripodoro, and Yero further asked whether it is true that $χ_μ(Q_{n})=O(1)$. In this note we answer their question in the negative by showing that $$ω(1)=χ_μ(Q_{n})=O(\log\log{n}).$$

math.CO↗
arXiv2026

Remarks on a theorem of Erdős and Szemerédi

Given a graph $G$ and a real $\varepsilon>0$, an edge-coloring of $G$ is called $\varepsilon$-balanced if each color appears on at least an $\varepsilon$-fraction of the edges in $G$. A classical result of Erdős and Szemerédi asserts that if a $2$-edge-coloring of a complete graph $K_n$ is not $\varepsilon$-balanced for some $0<\varepsilon\leq1/2$, then there exists a large monochromatic clique. This theorem has been used extensively in Ramsey-type arguments, as it allows one to focus on reasonably balanced colorings. However, in its original formulation the dependence between $n$ and $\varepsilon$ was left implicit, occasionally leading to inaccurate applications. In this short note, we revisit the Erdős--Szemerédi theorem and specify all parameter dependencies.

math.CO↗
arXiv2026

Induced saturation for complete bipartite posets

Given $s,t\in\mathbb{N}$, a complete bipartite poset $\mathcal{K}_{s,t}$ is a poset whose Hasse diagram consists of $s$ pairwise incomparable vertices in the upper layer and $t$ pairwise incomparable vertices in the lower layer, such that every vertex in the upper layer is larger than all vertices in the lower layer. A family $\mathcal{F}\subseteq2^{[n]}$ is called induced $\mathcal{K}_{s,t}$-saturated if $(\mathcal{F},\subseteq)$ contains no induced copy of $\mathcal{K}_{s,t}$, whereas adding any set from $2^{[n]}\backslash\mathcal{F}$ to $\mathcal{F}$ creates an induced $\mathcal{K}_{s,t}$. Let $\mathrm{sat}^{*}(n,\mathcal{K}_{s,t})$ denote the smallest size of an induced $\mathcal{K}_{s,t}$-saturated family $\mathcal{F}\subseteq2^{[n]}$. It was conjectured that $\mathrm{sat}^{*}(n,\mathcal{K}_{s,t})$ is superlinear in $n$ for certain values of $s$ and $t$. In this paper, we show that $\mathrm{sat}^{*}(n,\mathcal{K}_{s,t})=O(n)$ for all fixed $s,t\in\mathbb{N}$. Moreover, we prove a linear lower bound on $\mathrm{sat}^{*}(n,\mathcal{P})$ for a large class of posets $\mathcal{P}$, particularly for $\mathcal{K}_{s,2}$ with $s\in\mathbb{N}$.

math.CO↗
arXiv2026

On multiplicities of interpoint distances

Given a set $X\subseteq\mathbb{R}^2$ 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 $a_1(X) \geq a_2(X) \geq \cdots \geq a_m(X)$ denote the multiplicities of the $m$ distances determined by $X$ and let $a(X)=\left(a_1(X),\dots,a_m(X)\right)$. In this paper, we study several questions from Erdős's time regarding distance multiplicities. Among other results, we show that: (1) If $X$ is convex or ``not too convex'', then there exists a distance other than the diameter that has multiplicity at most $n$. (2) There exists a set $X \subseteq \mathbb{R}^2$ of $n$ points, such that many distances occur with high multiplicity. In particular, at least $n^{Ω(1/\log\log{n})}$ distances have superlinear multiplicity in $n$. (3) For any (not necessarily fixed) integer $1\leq k\leq\log{n}$, there exists $X\subseteq\mathbb{R}^2$ of $n$ points, such that the difference between the $k^{\text{th}}$ and $(k+1)^{\text{th}}$ largest multiplicities is at least $Ω(\frac{n\log{n}}{k})$. Moreover, the distances in $X$ with the largest $k$ multiplicities can be prescribed. (4) For every $n\in\mathbb{N}$, there exists $X\subseteq\mathbb{R}^2$ of $n$ points, not all collinear or cocircular, such that $a(X)= (n-1,n-2,\ldots,1)$. There also exists $Y\subseteq\mathbb{R}^2$ of $n$ points with pairwise distinct distance multiplicities and $a(Y) \neq (n-1,n-2,\ldots,1)$.

math.CO↗
arXiv2025

Ramsey problems for graphs in Euclidean spaces and Cartesian powers

Given a graph $H$, let $χ_H(\mathbb{R}^n)$ be the smallest positive integer $r$ such that there exists an $r$-coloring of $\mathbb{R}^n$ with no monochromatic unit-copy of $H$, that is a set of $|V(H)|$ vertices of the same color such that any two vertices corresponding to an edge of $H$ are at distance one. This Ramsey-type function extends the famous Hadwiger--Nelson problem on the chromatic number $χ(\mathbb{R}^n)=χ_{K_2}(\mathbb{R}^n)$ of the space from a complete graph $K_2$ on two vertices to an arbitrary graph $H$. It also extends the classical Euclidean Ramsey problem for congruent monochromatic subsets to the family of those defined by a specific subset of unit distances. Among others, we show that $χ_H(\mathbb{R}^n)=χ(\mathbb{R}^n)$ for any even cycle $H$ of length $8$ or at least $12$ as well as for any forest and that $χ_H(\mathbb{R}^n)=\lceilχ(\mathbb{R}^n)/2\rceil$ for any sufficiently long odd cycle. Our main tools and results, which are of independent interest, establish that Cartesian powers enjoy Ramsey-type properties for graphs with favorable Turán-type characteristics, such as zero hypercube Turán density. In addition, we prove induced variants of these results, find bounds on $χ_H(\mathbb{R}^n)$ for growing dimensions $n$, and prove a canonical-type result. We conclude with many open problems. One of these is to determine $χ_{C_4}(\mathbb{R}^2)$, for a cycle $C_4$ on four vertices.

math.CO↗
arXiv2025

On the number of sets with small sumset

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 $m \ll s^2/(\log n)^2$, then the number of subsets $A \subseteq Y$ with $|A| = s$ and $|A + A| \leq m$ is at most \[2^{o(s)}\binom{\frac{m+β}{2}}{s},\] where $β$ is the size of the largest subgroup of $G$ of size at most $\left(1+o(1)\right)m$. This bound is sharp for $\mathbb{Z}$ and many other groups. Our result improves the one of Campos and nearly bridges the remaining gap in a conjecture of Alon, Balogh, Morris, and Samotij. We also explore the behaviour of uniformly chosen random sets $A \subseteq \{1,\ldots,n\}$ with $|A| = s$ and $|A + A| \leq m$. Under the same assumption that $m \ll s^2/(\log n)^2$, we show that with high probability there exists an arithmetic progression $P \subseteq \mathbb{Z}$ of size at most $m/2 + o(m)$ containing all but $o(s)$ elements of $A$. Analogous results are obtained for asymmetric sumsets, improving results by Campos, Coulson, Serra, and Wötzel. The main tool behind our results is a more efficient container-type theorem developed for sets with small sumset, which gives an essentially optimal collection of containers. The proof of this combines an adapted hypergraph container lemma, that caters to the asymmetric setup as well, with a novel ``preprocessing'' graph container lemma, which allows the hypergraph container lemma to be called upon significantly less times than was necessary before.

math.CO↗
arXiv2022

The Turán number of the Cartesian product of graphs

Recently, Domagoj Bradač, Oliver Janzer, Benny Sudakov and István Tomon have proved that the Turán number of $2$-dimensional grids is $Θ(n^{3/2})$, or more general, $\mathrm{ex}\left(n,T\square{P}\right)=Θ(n^{3/2})$, where $T$ is a non-trivial tree, $P$ is a non-trivial path, and $T\square{P}$ denotes the Cartesian product. In their proof, they exhibited a novel way of using the tensor power trick, which has lots of potential in Turán type problems. By the end of their proof, they conjectured that $\mathrm{ex}\left(n,T\square{R}\right)=Θ(n^{3/2})$ for non-trivial trees $T$ and $R$. This paper is an extension based on their work, we successfully prove the above conjecture by adapting their approach.

math.CO↗