arXiv subjects

Ji Zeng

Publications and source records attributed to Ji Zeng.

34 records · Page 2Linked to original sources

arXiv2025

On higher dimensional point sets in general position

A finite point set in $\mathbb{R}^d$ is in general position if no $d + 1$ points lie on a common hyperplane. Let $α_d(N)$ be the largest integer such that any set of $N$ points in $\mathbb{R}^d$, with no $d + 2$ members on a common hyperplane, contains a subset of size $α_d(N)$ in general position. Using the method of hypergraph containers, Balogh and Solymosi showed that $α_2(N) < N^{5/6 + o(1)}$. In this paper, we also use the container method to obtain new upper bounds for $α_d(N)$ when $d \geq 3$. More precisely, we show that if $d$ is odd, then $α_d(N) < N^{\frac{1}{2} + \frac{1}{2d} + o(1)}$, and if $d$ is even, we have $α_d(N) < N^{\frac{1}{2} + \frac{1}{d-1} + o(1)}$. We also study the classical problem of determining $a(d,k,n)$, the maximum number of points selected from the grid $[n]^d$ such that no $k + 2$ members lie on a $k$-flat, and improve the previously best known bound for $a(d,k,n)$, due to Lefmann in 2008, by a polynomial factor when $k$ = 2 or 3 (mod 4).

math.CO↗

arXiv2025

Maximizing the Maximum Degree in Ordered Nearest Neighbor Graphs

For an ordered point set in a Euclidean space or, more generally, in an abstract metric space, the ordered Nearest Neighbor Graph is obtained by connecting each of the points to its closest predecessor by a directed edge. We show that for every set of $n$ points in $\mathbb{R}^d$, there exists an order such that the corresponding ordered Nearest Neighbor Graph has maximum degree at least $\log{n}/(4d)$. Apart from the $1/(4d)$ factor, this bound is the best possible. As for the abstract setting, we show that for every $n$-element metric space, there exists an order such that the corresponding ordered Nearest Neighbor Graph has maximum degree $Ω(\sqrt{\log{n}/\log\log{n}})$.

math.CO↗

arXiv2025

Saturation results around the Erdős--Szekeres problem

In this paper, we consider saturation problems related to the celebrated Erdős--Szekeres convex polygon problem. For each $n \ge 7$, we construct a planar point set of size $(7/8) \cdot 2^{n-2}$ which is saturated for convex $n$-gons. That is, the set contains no $n$ points in convex position while the addition of any new point creates such a configuration. This demonstrates that the saturation number is smaller than the Ramsey number for the Erdős--Szekeres problem. The proof also shows that the original Erdős--Szekeres construction is indeed saturated. Our construction is based on a similar improvement for the saturation version of the cups-versus-caps theorem. Moreover, we consider the generalization of the cups-versus-caps theorem to monotone paths in ordered hypergraphs. In contrast to the geometric setting, we show that this abstract saturation number is always equal to the corresponding Ramsey number.

math.CO↗

arXiv2025

LIMI: Less is More for Agency

We define Agency as the emergent capacity of AI systems to function as autonomous agents actively discovering problems, formulating hypotheses, and executing solutions through self-directed engagement with environments and tools. This fundamental capability marks the dawn of the Age of AI Agency, driven by a critical industry shift: the urgent need for AI systems that don't just think, but work. While current AI excels at reasoning and generating responses, industries demand autonomous agents that can execute tasks, operate tools, and drive real-world outcomes. As agentic intelligence becomes the defining characteristic separating cognitive systems from productive workers, efficiently cultivating machine autonomy becomes paramount. Current approaches assume that more data yields better agency, following traditional scaling laws from language modeling. We fundamentally challenge this paradigm. LIMI (Less Is More for Intelligent Agency) demonstrates that agency follows radically different development principles. Through strategic focus on collaborative software development and scientific research workflows, we show that sophisticated agentic intelligence can emerge from minimal but strategically curated demonstrations of autonomous behavior. Using only 78 carefully designed training samples, LIMI achieves 73.5% on comprehensive agency benchmarks, dramatically outperforming state-of-the-art models: Kimi-K2-Instruct (24.1%), DeepSeek-V3.1 (11.9%), Qwen3-235B-A22B-Instruct (27.5%), and GLM-4.5 (45.1%). Most strikingly, LIMI demonstrates 53.7% improvement over models trained on 10,000 samples-achieving superior agentic intelligence with 128 times fewer samples. Our findings establish the Agency Efficiency Principle: machine autonomy emerges not from data abundance but from strategic curation of high-quality agentic demonstrations.

cs.AI↗

arXiv2025

A Purely Geometric Variant of the Gale-Berlekamp Switching Game

We introduce the following variant of the Gale-Berlekamp switching game. Let $P$ be a set of n noncollinear points in the plane, each of them having weight $+1$ or $-1$. At each step, we pick a line $\ell$ passing through at least two points of $P$, and switch the sign of every point $p \in P\cap\ell$. The objective is to maximize the total weight of the elements of $P$. We show that one can always achieve that this quantity is at least $n - o(n)$, as $n\rightarrow\infty$, and at least $n/3$, for every $n$. Moreover, these can be attained by a polynomial time algorithm.

cs.CG↗

arXiv2025

Evasive sets, twisted varieties, and container-clique trees

In the affine space $\mathbb{F}_q^n$ over the finite field of order $q$, a point set $S$ is said to be $(d,k,r)$-evasive if the intersection between $S$ and any variety, of dimension $k$ and degree at most $d$, has cardinality less than $r$. As $q$ tends to infinity, the size of a $(d,k,r)$-evasive set in $\mathbb{F}_q^n$ is at most $O\left(q^{n-k}\right)$ by a simple averaging argument. We exhibit the existence of such evasive sets of sizes at least $Ω\left(q^{n-k}\right)$ for much smaller values of $r$ than previously known constructions, and establish an enumerative upper bound $2^{O(q^{n-k})}$ for the total number of such evasive sets. The existence result is based on our study of twisted varieties. In the projective space $\mathbb{P}^n$ over an algebraically closed field, a variety $V$ is said to be $d$-twisted if the intersection between $V$ and any variety, of dimension $n - \dim(V)$ and degree at most $d$, has dimension zero. We prove an upper bound on the smallest possible degree of twisted varieties which is best possible in a mild sense. The enumeration result includes a new technique for the container method which we believe is of independent interest. To illustrate the potential of this technique, we give a simpler proof of a result by Chen--Liu--Nie--Zeng that characterizes the maximum size of a collinear-triple-free subset in a random sampling of $ \mathbb{F}_q^2$ up to polylogarithmic factors.

math.CO↗

arXiv2025

Unbalanced Zarankiewicz problem for bipartite subdivisions with applications to incidence geometry

For a bipartite graph $H$, its linear threshold is the smallest real number $σ$ such that every bipartite graph $G = (U \sqcup V, E)$ with unbalanced parts $|V| \gtrsim |U|^σ$ and without a copy of $H$ must have a linear number of edges $|E| \lesssim |V|$. We prove that the linear threshold of the complete bipartite subdivision graph $K_{s,t}'$ is at most $σ_s = 2 - 1/s$. Moreover, we show that any $σ< σ_s$ is less than the linear threshold of $K_{s,t}'$ for sufficiently large $t$ (depending on $s$ and $σ$). Some geometric applications of this result are given: we show that any $n$ points and $n$ lines in the complex plane without an $s$-by-$s$ grid determine $O(n^{4/3 - c})$ incidences for some constant $c > 0$ depending on $s$; and for certain pairs $(p,q)$, we establish nontrivial lower bounds on the number of distinct distances determined by $n$ points in the plane under the condition that every $p$ points determine at least $q$ distinct distances.

math.CO↗

arXiv2025

Random Turán and counting results for general position sets over finite fields

Let $α(\mathbb{F}_q^d,p)$ denote the maximum size of a general position set in a $p$-random subset of $\mathbb{F}_q^d$. We determine the order of magnitude of $α(\mathbb{F}_q^2,p)$ up to polylogarithmic factors for all possible values of $p$, improving the previous results obtained by Roche-Newton--Warren and Bhowmick--Roche-Newton. For $d \ge 3$ we prove upper bounds for $α(\mathbb{F}_q^d,p)$ that are essentially tight within certain ranges for $p$. We establish the upper bound $2^{(1+o(1))q}$ for the number of general position sets in $\mathbb{F}_q^d$, which matches the trivial lower bound $2^{q}$ asymptotically in the exponent. We also refine this counting result by proving an asymptotically tight (in the exponent) upper bound for the number of general position sets with a fixed size. The latter result for $d=2$ improves a result of Roche-Newton--Warren. Our proofs are grounded in the hypergraph container method, and additionally, for $d=2$ we also leverage the pseudorandomness of the point-line incidence graph of $\mathbb{F}_{q}^2$.

math.CO↗

arXiv2024

Note on disjoint faces in simple topological graphs

We prove that every $n$-vertex complete simple topological graph generates at least $Ω(n)$ pairwise disjoint $4$-faces. This improves upon a recent result by Hubard and Suk. As an immediate corollary, every $n$-vertex complete simple topological graph drawn in the unit square generates a $4$-face with area at most $O(1/n)$. This can be seen as a topological variant of the Heilbronn problem for quadrilaterals. We construct examples showing that our result is asymptotically tight. We also discuss the similar problem for $k$-faces with arbitrary $k\geq 3$.

math.CO↗

arXiv2024

3-uniform monotone paths and multicolor Ramsey numbers

The monotone path $P_{n+2}$ is an ordered 3-uniform hypergraph whose vertex set has size $n+2$ and edge set consists of all consecutive triples. In this note, we consider the collection $\mathcal{J}_n$ of ordered 3-uniform hypergraphs named monotone paths with $n$ jumps, and we prove the following relation \begin{equation*} r(3;n) \leq R(P_{n+2},\mathcal{J}_n) \leq 4^n \cdot r(3;n), \end{equation*} where $r(3;n)$ is the multicolor Ramsey number for triangles and $R(P_{n+2},\mathcal{J}_n)$ is the hypergraph Ramsey number for $P_{n+2}$ versus any member of $\mathcal{J}_n$. In particular, whether $r(3;n)$ is exponential, which is a very old problem of Erdős, is equivalent to whether $R(P_{n+2},\mathcal{J}_n)$ is exponential.

math.CO↗

arXiv2024

On cliques in three-dimensional dense point-line arrangements

As a variant of the celebrated Szemerédi--Trotter theorem, Guth and Katz proved that $m$ points and $n$ lines in $\mathbb{R}^3$ with at most $\sqrt{n}$ lines in a common plane must determine at most $O(m^{1/2}n^{3/4})$ incidences for $n^{1/2}\leq m\leq n^{3/2}$. This upper bound is asymptotically tight and has an important application in Erdős distinct distance problem. We characterize the extremal constructions towards the Guth--Katz bound by proving that such a large dense point-line arrangement must contain a $k$-clique in general position provided $m \ll n$. This is an analog of a result by Solymosi for extremal Szemerédi--Trotter constructions in the plane.

math.CO↗

arXiv2023

A positive fraction Erdos-Szekeres theorem and its applications

A famous theorem of Erdos and Szekeres states that any sequence of $n$ distinct real numbers contains a monotone subsequence of length at least $\sqrt{n}$. Here, we prove a positive fraction version of this theorem. For $n > (k-1)^2$, any sequence $A$ of $n$ distinct real numbers contains a collection of subsets $A_1,\ldots, A_k \subset A$, appearing sequentially, all of size $s=Ω(n/k^2)$, such that every subsequence $(a_1,\ldots, a_k)$, with $a_i \in A_i$, is increasing, or every such subsequence is decreasing. The subsequence $S = (A_1,\ldots, A_k)$ described above is called block-monotone of depth $k$ and block-size $s$. Our theorem is asymptotically best possible and follows from a more general Ramsey-type result for monotone paths, which we find of independent interest. We also show that for any positive integer $k$, any finite sequence of distinct real numbers can be partitioned into $O(k^2\log k)$ block-monotone subsequences of depth at least $k$, upon deleting at most $(k-1)^2$ entries. We apply our results to mutually avoiding planar point sets and biarc diagrams in graph drawing.

math.CO↗

arXiv2022

Unavoidable patterns in complete simple topological graphs

We show that every complete $n$-vertex simple topological graph contains a topological subgraph on at least $(\log n)^{1/4 - o(1)}$ vertices that is weakly isomorphic to the complete convex geometric graph or the complete twisted graph. This is the first improvement on the bound $Ω(\log^{1/8}n)$ obtained in 2003 by Pach, Solymosi, and Tóth. We also show that every complete $n$-vertex simple topological graph contains a plane path of length at least $(\log n)^{1 -o(1)}$.

math.CO↗

arXiv2022

Forest formulas of discrete Green's functions

The discrete Green's functions are the pseudoinverse (or the inverse) of the Laplacian (or its variations) of a graph. In this paper, we will give combinatorial interpretations of Green's functions in terms of enumerating trees and forests in a graph that will be used to derive further formulas for several graph invariants. For example, we show that the trace of the Green's function $\mathbf{G}$ associated with the combinatorial Laplacian of a connected simple graph $Γ$ on $n$ vertices satisfies $\text{Tr}(\mathbf{G})=\sum_{λ_i \neq 0} \frac 1 {λ_i}= \frac{1}{nτ}|\mathbb{F}^*_2|$, where $λ_i$ denotes the eigenvalues of the combinatorial Laplacian, $τ$ denotes the number of spanning trees and $\mathbb{F}^*_2$ denotes the set of rooted spanning $2$-forests in $Γ$. We will prove forest formulas for discrete Green's functions for directed and weighted graphs and apply them to study random walks on graphs and digraphs. We derive a forest expression of the hitting time for digraphs, which gives combinatorial proofs to old and new results about hitting times, traces of discrete Green's functions, and other related quantities.

math.CO↗

arXiv2022

On asymptotic packing of convex geometric and ordered graphs

A convex geometric graph $G$ is said to be packable if there exist edge-disjoint copies of $G$ in the complete convex geometric graph $K_n$ covering all but $o(n^2)$ edges. We prove that every convex geometric graph with cyclic chromatic number at most $4$ is packable. With a similar definition of packability for ordered graphs, we prove that every ordered graph with interval chromatic number at most $3$ is packable. Arguments based on the average length of edges imply these results are best possible. We also identify a class of convex geometric graphs that are packable due to having many "long" edges.

math.CO↗

arXiv2022

Semi-restricted Rock, Paper, Scissors

Consider the following variant of Rock, Paper, Scissors (RPS) played by two players Rei and Norman. The game consists of $3n$ rounds of RPS, with the twist being that Rei (the restricted player) must use each of Rock, Paper, and Scissors exactly $n$ times during the $3n$ rounds, while Norman is allowed to play normally without any restrictions. Answering a question of Spiro, we show that a certain greedy strategy is the unique optimal strategy for Rei in this game, and that Norman's expected score is $Θ(\sqrt{n})$. Moreover, we study semi-restricted versions of general zero sum games and prove a number of results concerning their optimal strategies and expected scores, which in particular implies our results for semi-restricted RPS.

math.CO↗