arXiv subjects

Zixiang Xu

Publications and source records attributed to Zixiang Xu.

At least 19 recordsLinked to original sources

arXiv2026

AlgoWorlds: Benchmarking Tool Use for Global Optimization in Algorithmic Worlds

Tool-use benchmarks generally evaluate whether an agent completes a workflow using appropriate tools and valid arguments. However, feasibility alone is insufficient in real-world decision settings such as route planning and fleet dispatch. Individual choices interact through shared constraints and costs, so a feasible solution may still be substantially suboptimal. This raises a harder question: can an agent turn information gathered through tools into a globally optimal decision? We introduce AlgoWorlds, a benchmark that transforms formally specified combinatorial optimization problems into partially observed decision environments with verifiable global optima. Each environment contains a hidden instance observed only through task-specific information tools, after which the agent commits to one structured decision evaluated for feasibility and optimality. AlgoWorlds contains 240 environments covering ten combinatorial optimization families and four workload levels. Family-specific deterministic programs generate the instances, exact algorithms certify their optima and determine workload levels, and two structurally different tool interfaces present each underlying instance. We evaluate seven leading LLMs, including Claude Opus 4.8 and GPT-5.6 Sol. Achieving global optimality remains highly challenging: although leading models produce feasible decisions in most cases, the best-performing model reaches exact optimality in only 38.61% of cases. Even when agents collect sufficient information to reconstruct the hidden instance, most failures end in feasible but suboptimal decisions. The challenge therefore extends beyond information acquisition to information integration, global constraint reasoning, and decision verification. The project homepage is available at https://xzx34.github.io/AlgoWorlds/, and the code is available at https://github.com/xzx34/AlgoWorlds.

cs.CL↗

arXiv2026

Economical lattice coverings by determined segments

For fixed $d\geq 2$, let $\tau_d(n)$ be the minimum size of a set $S\subseteq\{0,\ldots,n\}^d$ such that the affine lines determined by pairs of distinct points of $S$ cover the grid. Let $\sigma_d(n)$ be the analogous minimum when every grid point must lie on the closed segment joining two distinct points of $S$. A celebrated result of Alon [GAFA, 1991] proved that $\tau_d(n)$ is of order between $\Omega_d(n^{\alpha_d})$ and $O_d(n^{\alpha_d}\log n)$, where $\alpha_d=\frac{d(d-1)}{2d-1}$, and asked whether the logarithm term is necessary. We prove that $$c_d n^{\alpha_d}\leq\tau_d(n)\leq\sigma_d(n)\leq C_d n^{\alpha_d}$$ for every fixed $d\geq 2$, thereby resolving Alon's problem in a stronger form.

math.CO↗

arXiv2026

Boundary-Induced Apparent Risk Aversion in Nonergodic Multiplicative Growth

Observed risk-taking behavior is often rationalized through expected-utility curvature, yet the curvature required to fit choices in one context can differ sharply from the curvature required in another, a tension highlighted by calibration critiques of expected-utility theory. Finite multiplicative systems often cease to evolve when a lower continuation threshold is reached, whereas standard growth-optimal benchmarks assume uninterrupted continuation. We study a finite-horizon binary multiplicative process in which a fixed exposure is chosen ex ante and paths crossing an absorbing boundary are assigned a residual value. Exact lattice propagation yields the optimal exposure as a function of initial log distance to the boundary, horizon, and residual ratio. Costly absorption compresses exposure below the no-boundary Kelly fraction near the boundary. When interpreted through an unconstrained constant-relative-risk-aversion benchmark, this compression appears as elevated risk aversion. As the residual value approaches the boundary, a local above-Kelly reversal can occur. In this minimal finite-horizon setting, absorbing-boundary geometry is therefore sufficient to generate state-dependent risk-averse-looking behavior without heterogeneous primitive preference parameters.

econ.GN↗

arXiv2026

Largest Sperner families with restricted differences

Let $L$ be a fixed set of positive integers. A family $\mathcal{F}\subseteq 2^{[n]}$ is called $L$-differencing if $\lvert A\setminus B\rvert\in L$ for every ordered pair of distinct members $A,B\in\mathcal{F}$. A longstanding conjecture of Frankl, proposed in 1985, asserts that every $L$-differencing family has size at most $\binom{n}{|L|}$. We resolve this conjecture asymptotically for every fixed $L$, and obtain the exact answer in the only case in which the conjectured bound could be tight. (1) If $L\ne [s]$ and $n$ is large, then every $L$-differencing family satisfies $\lvert \mathcal{F}\rvert \le \left(\frac{s}{s+1}+o_L(1)\right)\binom{n}{s}$. (2) If $L=[s]$ and $n\ge 2s-1$, then $\lvert \mathcal{F}\rvert\le\binom{n}{s}$, with equality only for $\binom{[n]}{s}$ and $\binom{[n]}{n-s}$. The first result follows by reducing directed differences to restricted Hamming distances. For the exact result, we develop a new homogeneous polynomial method, which might be of independent interest.

math.CO↗

arXiv2026

Tverberg's theorem for unions of convex sets: Sharp bounds and colored extensions

Let $f_{r}(d,s_{1},\ldots,s_{r})$ be the least $N$ such that every $N$-point set $P\subseteq\mathbb{R}^{d}$ has an $r$-partition $P=P_{1}\sqcup\cdots\sqcup P_{r}$ with the following property: whenever $C_{i}\supseteq P_{i}$ is a union of at most $s_{i}$ convex sets, one has $\bigcap_{i=1}^{r}C_{i}\ne\emptyset$. A recent breakthrough of Alon and Smorodinsky proved that $f_{r}(d,s,\ldots,s)\le cdr^{2}s^{r}\log r\log(es^{r})$ for an absolute constant $c>0$. In this paper, we determine the asymptotic order in two principal ranges: $f_{2}(2,s,s)=\Theta(s^{2})$, and $f_{r}(d,s,\ldots,s)=\Theta_{d,r}(s^{r}\log{s})$ for every fixed $d,r$ with $d\ge r+2$. The first one determines the order of the extremal function proposed by Kalai from the 1970s. Together, the two results show a sharp dependence on the dimension: for two parts, the logarithmic factor disappears in the plane but is necessary in every fixed dimension $d\ge4$. Beyond these sharp results, when $r\ge d+1$ we improve the upper bound of Alon and Smorodinsky by proving both $f_{r}(d,s,\ldots,s)\le c_{d}rs^{r}\log(ers^{r})$ and $f_{r}(d,s,\ldots,s)\le c_{d}r^{d+2}s^{d+1}\log(ers)$ through a local Helly-type argument. We also prove $f_{r}(d,s,\ldots,s)>s^{\min\{r,d\}}$ for every $d\ge2$. Finally, we study two colored analogues. The direct B\'{a}r\'{a}ny--Larman-type extension, in which one seeks $r$ disjoint rainbow sets chosen from $d+1$ color classes, fails as soon as two convex pieces are allowed. Nevertheless, a different extension does hold: given sufficiently many prescribed $r$-point classes, one can split every class completely among the $r$ final parts while retaining the required intersection property.

math.CO↗

arXiv2026

All simplices exhibit canonical Ramsey property

We prove that all nondegenerate simplices have the canonical Ramsey property, thereby resolving a central open problem in canonical Euclidean Ramsey theory and providing a canonical counterpart to the celebrated simplex Ramsey theorem of Frankl and R\"{o}dl~[JAMS, 1990].

math.CO↗

arXiv2026

Inside the Unfair Judge: A Mechanistic Interpretability Account of LLM-as-Judge Bias

Existing studies of LLM-as-judge scoring bias work predominantly at the input-output level: they perturb inputs, measure score deltas, and propose prompt-level mitigations. We argue that the same biases admit a representation-level account in the judge's hidden state, complementary to the input-output view and operationally useful in ways it does not afford. We report three findings, across seven judges, seven bias types, and nine benchmarks. Geometry: baseline judging inputs occupy a tight activation manifold while biased inputs are displaced along a low-dimensional, type-specific subspace that sharpens with depth and is recovered consistently by three families of estimators. Causal control: steering hidden states along this subspace drives scoring in both directions, forward shifts reproducing biased scoring on clean inputs and reverse shifts restoring baseline scoring on biased ones, while matched-norm random directions produce shifts an order of magnitude smaller. Operational: a simple linear projection onto the same bias-direction features anticipates judge failures on three entirely unseen benchmarks, substantially outperforming text-based alternatives. Reading bias as activation geometry, rather than as input-output noise, unifies geometric structure, causal control, and operational prediction within a single framework. The project page is available at https://xzx34.github.io/unfair-judge/

cs.LG↗

arXiv2026

The sharp asymptotic density of zero-sum-free spherical sets

A measurable set $A\subseteq \mathbb S^{d-1}$ is called zero-sum-free if there are no $\boldsymbol{x},\boldsymbol{y},\boldsymbol{z}\in A$ with $\boldsymbol{x}+\boldsymbol{y}+\boldsymbol{z}=\boldsymbol{0}$. Bukh asked whether every zero-sum-free measurable subset of $\mathbb S^{d-1}$, for $d\ge3$, has normalized surface measure at most $\frac{1}{2}$. He also pointed out that even the asymptotic behavior as $d\to\infty$ was unknown. We answer Bukh's asymptotic question by proving that every such set has normalized surface measure at most $\frac{\lfloor (d+1)^2/2\rfloor}{d(d+1)}=\frac{1}{2}+O\left(\frac{1}{d}\right).$ Since the lower bound $\frac{1}{2}$ comes from open hemispheres, this determines the asymptotic extremal density. By monotonicity, upper bounds in low-dimensional cases are especially important. We use a stability argument to improve the bound from $\frac{3}{5}$ to $\frac{71}{120}$ in dimensions $4$ and $5$.

math.CO↗

arXiv2026

A disproof of the uniform witness conjecture

The study of $(d+1)$-uniform set systems with VC-dimension at most $d$ links the Erd\H{o}s--Ko--Rado theorem with VC-dimension. But already in 1997, Ahlswede and Khachatrian showed that this is not the right extension of the Erd\H{o}s--Ko--Rado theorem. In 2025, Chao, Xu, Yip and Zhang proposed the uniform witness conjecture as a possible right extension: for $0\le s\le d$, if every set of a $(d+1)$-uniform family has a missing trace of the same fixed size $s$, then the family should have size at most $\binom{n-1}{d}$. They proved the conjecture when $s=d$, and when $s=1$ and $n$ is large. Very recently, Chao, Xu and Zakharov proved the conjecture when $s\le \frac{d}{2}$ and $n$ is large. We fill in the missing half of the picture, although the picture is not the one suggested by the conjecture. More precisely, for $d\ge 4$ and $\left\lceil \frac{d+2}{2}\right\rceil\le s\le d-1$, we construct such a family $\mathcal{F}\subseteq\binom{[n]}{d+1}$ with $|\mathcal{F}|=\binom{n-1}{d}+\binom{n-2(d+1-s)-2}{2s-d-2}$ for every $n\ge2(d+1)$, thereby disproving the uniform witness conjecture.

math.CO↗

arXiv2026

Beating the Ahlswede--Khachatrian bound for the Erd\H{o}s--Frankl--Pach problem

In the 1980s, Erd\H{o}s and, independently, Frankl and Pach conjectured that, for sufficiently large $n$, every $(d+1)$-uniform family on $\{1,\ldots,n\}$ with VC-dimension $d$ has size at most $\binom{n-1}{d}$, the size of a star. Ahlswede and Khachatrian disproved this conjecture in 1997 by giving a family of size $\binom{n-1}{d}+\binom{n-4}{d-2}$. This value has since been widely believed to be best possible, and Mubayi and Zhao explicitly conjectured its optimality in 2007. Very recently, Wang, Xu and Zhang proved their conjecture for $d=2$ and $n\ge 7$, providing further support for this belief. Surprisingly, we show that the Mubayi-Zhao conjecture is false for every $d\ge 3$ by constructing families larger than the Ahlswede--Khachatrian bound. Our constructions suggest that the answer to the Erd\H{o}s--Frankl--Pach problem depends delicately on both $n$ and $d$.

math.CO↗

arXiv2026

Balanced Sperner families via the topological Tverberg theorem

For every prime power $r\ge 2$, we show that any Sperner family $\mathcal F\subseteq 2^{[n]}$ with $|\mathcal F|\ge (r-1)n+1$ contains $r$ pairwise disjoint nonempty subfamilies whose unions are all equal and whose intersections are all equal. For $r=2$, this confirms a conjecture of Heged\"{u}s, with the sharp threshold $n+1$. In this purely combinatorial problem, our proof combines a multilinear polynomial method, a continuity argument, and the topological Tverberg theorem.

math.CO↗

arXiv2026

Kusner's conjecture: Exact values and linear bounds

In 1983, Kusner conjectured that the largest equilateral set in $\mathbb{R}^{n}$ with metric $\ell_{p}$ has cardinality $n+1$ when $1 2$, improving Alon's bounds $O_p(n^{2+\frac{2}{\lfloor p\rfloor}})$ for all finite $p\ge 1$.

math.CO↗

arXiv2026

Kleitman's theorem over vector spaces: parity phenomena in canonical and global stability

In 1966, Kleitman determined the maximum size of a family of subsets of $[n]$ with bounded symmetric difference. Liao, Liu and Yan recently established a vector-space analogue in the cases $n=d+1$ and $n>2d$, and asked for the sharp bound in the remaining range. We resolve this problem completely by proving the exact vector-space analogue of Kleitman's theorem for every $n\ge d+1$, and we also determine all extremal configurations. We further develop a stability theory for the vector-space diameter problem. Unlike the Boolean cube, the lattice of subspaces has no translation symmetry, and this makes the stability theory substantially different from its classical counterpart. The geometry of subspace balls leads to two natural notions: canonical stability, which forbids containment only in the canonical extremal configurations, and global stability, which forbids containment in arbitrary balls or adjacent double balls of the corresponding radius. We determine sharp canonical stability in even diameter, sharp canonical and global stability in odd diameter, and prove a nontrivial general upper bound for global stability in even diameter. In particular, these two notions exhibit a sharp parity split: in odd diameter they collapse to the same problem, whereas in even diameter they lead to genuinely different extremal behavior.

math.CO↗

arXiv2026

A solution to a strengthened conjecture of Bukh, van Hintum and Keevash on additive bases

Motivated by the change-of-domain problem for additive bases, Bukh, van Hintum and Keevash conjectured that if $A,B\subseteq \mathbb{Q}^{n}$ and $\{\boldsymbol{e}_i+\boldsymbol{e}_j:1\le i\le j\le n\}\subseteq A+B,$ then $|A|+|B|\ge 2n$. They further proposed the strengthened conjecture: if $|A|=n-t$, then $|B|\ge n+\binom{t+1}{2}.$ Bukh also explicitly asked whether the same bounds hold for $A,B\subseteq \mathbb{R}^{n}$ and an arbitrary basis $S$ of $\mathbb{R}^{n}$, under the assumption $S+S\subseteq A+B$. We prove the full strengthened statement over $\mathbb{R}^{n}$: if $S+S\subseteq A+B$ and $|A|\le n-t$ with $0\le t\le n-1$, then $|B|\ge n+\binom{t+1}{2},$ which is sharp for every basis $S$ and every $0\le t\le n-1.$ The proof is short, using edge contractions in a graph-theoretical framework and a new coloring lemma over $\mathbb F_2^n$.

math.CO↗

arXiv2026

Improvement on the Erdős-Kleitman conjecture via the KKL theorem

In 1974, Erdős and Kleitman conjectured that if a family $\mathcal{F}\subseteq 2^{[n]}$ contains no matching of size $s$ and is maximal with respect to this property, then $ |\mathcal{F}|\ge \left(1-2^{-(s-1)}\right)\cdot 2^{n}. $ For decades, the best general lower bound remained the trivial $2^{n-1}$. About a decade ago, Frankl and Tokushige emphasized that obtaining a bound of the form $\left(\frac{1}{2}+\varepsilon\right)\cdot 2^n$ for some $\varepsilon>0$ is a challenging problem. A breakthrough of Bucič, Letzter, Sudakov and Tran in 2018 showed that $ |\mathcal{F}|\ge \left(1-\frac{1}{s}\right)\cdot 2^n $ via two very elegant and quite different approaches. Our main result shows that $$ |\mathcal{F}|\ge \left( 1 - \frac{1}{s + (s-2)\frac{\log n}{2\sqrt{5}n}} \right)\cdot 2^n $$ by exploiting a connection to the cornerstone result of Kahn, Kalai and Linial on influences of Boolean functions. Independently, we can also obtain a weaker improvement combining the linear algebra method with a combinatorial twist.

math.CO↗

arXiv2026

Tur\'{a}n problems for multilinear maps

We study Tur\'{a}n-type extremal problems for alternating and unrestricted multilinear maps. For alternating order-$d$ multilinear maps $T: (\mathbb F^n)^d\to \mathbb{F}^m$, we determine, over algebraically closed fields of arbitrary characteristic, the largest $k$ such that every $T$ vanishes identically on $\mathbb{V}^d$ for some $k$-dimensional subspace $\mathbb{V}$. This extends the bilinear formula of Buhler, Gupta, and Harris [J. Algebra, 1987] to arbitrary order and resolves a question of Qiao [Discrete Anal., 2023]. We also solve the analogous problem for arbitrary, not necessarily alternating, multilinear maps by determining the largest $k$ such that every $T$ vanishes on $\mathbb{V}_1\times\cdots\times \mathbb{V}_d$ for some $k$-dimensional subspaces $\mathbb{V}_1,\dots,\mathbb{V}_d$. These results yield exact values, over algebraically closed fields, of the Feldman--Propp number [Adv. Math., 1992], the Tur\'{a}n number [Discrete Anal., 2023], and the Gow--Quinlan number [Linear Multilinear Algebra, 2006] associated with alternating multilinear maps. Finally, motivated by the Erd\H{o}s box problem, we give a purely algebraic derivation of the Conlon--Pohoata--Zakharov lower bound [Discrete Anal., 2021] by combining analytic and partition rank estimates with an incidence count. In the relevant parameter range, we further show that every multilinear map defined over a finite field has many isotropic tuples of $2$-dimensional subspaces over extensions of sufficiently divisible degree. This rules out the natural route to improving the Conlon--Pohoata--Zakharov exponent by selecting multilinear maps with substantially fewer bad isotropic configurations.

math.CO↗

arXiv2026

ERNIE 5.0 Technical Report

In this report, we introduce ERNIE 5.0, a natively autoregressive foundation model desinged for unified multimodal understanding and generation across text, image, video, and audio. All modalities are trained from scratch under a unified next-group-of-tokens prediction objective, based on an ultra-sparse mixture-of-experts (MoE) architecture with modality-agnostic expert routing. To address practical challenges in large-scale deployment under diverse resource constraints, ERNIE 5.0 adopts a novel elastic training paradigm. Within a single pre-training run, the model learns a family of sub-models with varying depths, expert capacities, and routing sparsity, enabling flexible trade-offs among performance, model size, and inference latency in memory- or time-constrained scenarios. Moreover, we systematically address the challenges of scaling reinforcement learning to unified foundation models, thereby guaranteeing efficient and stable post-training under ultra-sparse MoE architectures and diverse multimodal settings. Extensive experiments demonstrate that ERNIE 5.0 achieves strong and balanced performance across multiple modalities. To the best of our knowledge, among publicly disclosed models, ERNIE 5.0 represents the first production-scale realization of a trillion-parameter unified autoregressive model that supports both multimodal understanding and generation. To facilitate further research, we present detailed visualizations of modality-agnostic expert routing in the unified model, alongside comprehensive empirical analysis of elastic training, aiming to offer profound insights to the community.

cs.CL↗

arXiv2025

A Tverberg-type problem of Kalai: Two negative answers to questions of Alon and Smorodinsky, and the power of disjointness

Let $f_r(d,s_1,\ldots,s_r)$ denote the least integer $n$ such that every $n$-point set $P\subseteq\mathbb{R}^d$ admits a partition $P=P_1\cup\cdots\cup P_r$ with the property that for any choice of $s_i$-convex sets $C_i\supseteq P_i$ $(i\in[r])$ one necessarily has $\bigcap_{i=1}^r C_i\neq\emptyset$, where an $s_i$-convex set means a union of $s_i$ convex sets. A recent breakthrough by Alon and Smorodinsky establishes a general upper bound $f_r(d,s_1,\dots,s_r) = O(dr^2\log r \prod_{i=1}^r s_i\cdot \log(\prod_{i=1}^r s_i).$ Specializing to $r=2$ resolves the problem of Kalai from the 1970s. They further singled out two particularly intriguing questions: whether $f_{2}(2,s,s)$ can be improved from $O(s^2\log s)$ to $O(s)$, and whether $f_r(d,s,\ldots,s)\le Poly(r,d,s)$. We answer both in the negative by showing the exponential lower bound $f_{r}(d,s,\ldots,s)> s^{r}$ for any $r\ge 2$, $s\ge 1$ and $d\ge 2r-2$, which matches the upper bound up to a multiplicative $\log{s}$ factor for sufficiently large $s$. Our construction combines a scalloped planar configuration with a direct product of regular $s$-gon on the high-dimensional torus $(\mathbb{S}^1)^{r-2}$. Perhaps surprisingly, if we additionally require that within each block the $s_i$ convex sets are pairwise disjoint, the picture changes markedly. Let $F_r(d,s_1,\ldots,s_r)$ denote this disjoint-union variant of the extremal function. We show: (1) $F_{2}(2,s,s)=O(s\log s)$ by connecting it to a suitable line-separating function in the plane; (2) when $s$ is large, $F_r(d,s,\ldots,s)$ can be bounded by $O_{r,d}(s^{(1-\frac{1}{2^{d}(d+1)})r+1})$ and $O_{d}(r^{3}\log r\cdot s^{2d+3})$, respectively. This builds on a novel connection between the geometric obstruction and hypergraph Turán numbers, in particular, a variant of the Erdős box problem.

math.CO↗