Eleon Bach, Yann Disser, Sophie Huiberts, Nils Mosis
We prove that, for every fixed $ε\in(0,1]$, the active-set method requires exponentially many iterations in the worst case to maximize a convex objective of the form $(x_1)^{1+ε}+\vec c^{\top}\vec x$ subject to linear constraints, regardless of the pivot rule used. Our lower bound therefore holds for objectives arbitrarily close to the linear case $ε=0$, where the active-set method coincides with the simplex method and a lower bound for all pivot rules would constitute a major breakthrough. The special case $ε=1$ yields a lower bound in convex quadratic maximization. This improves on the best previously known lower bound [IPCO 2025] requiring polynomial objectives of degree $ω(\log d)$ in dimension $d$ and firmly resolves the open question of whether a constant degree suffices. Our result is based on a novel extended formulation, recursively constructed using deformed products. Its key feature is that it projects onto a polygonal approximation of a parabola while preserving all of its exponentially many vertices. We define a convex objective with a single non-linear variable that forces the active-set method to follow the parabolic boundary of this projection, without allowing any shortcuts along chords corresponding to edges of its full-dimensional preimage.
The spatial chromatic model (SCM) is a novel entity-oriented spatial data model proposed in recent years. Spatial tessellations formed by SCM have been applied in preliminary analyses of spatial point pattern recognition,spatial topology, and Voronoi diagram generation. Unlike previous spatial tessellation models, a key feature of SCM is that each subspace (cell) possesses a chromatic code, which serves as the foundation for various theoretical analyses and applications of SCM. The uniqueness of cell cluster codes in SCM constitutes an important property of the model. Previous studies have demonstrated the uniqueness of the chromatic codes of 2-cell and 3-cell connected disjoint clusters. Building on this analysis of connected cluster code characteristics, this paper proves the uniqueness of the chromatic codes of 4-cell connected disjoint clusters. The proof results and the new concepts introduced in this study help further refine the spatial chromatic model and provide a theoretical foundation for its subsequent applications.
We analyze the expected running time of WalkSAT, a well-known local search procedure for satisfiability solving, on satisfiable instances of the k-XOR SAT problem. We obtain estimates of this expected running time by reducing the problem to a setting amenable to classical techniques from drift analysis. A crucial ingredient of this reduction is the definition of (new, explosive) hypergraph versions of interacting particle systems, notably of coalescing and annihilating random walks as well as the voter model. The use of these tools allows to show that the expected running time of WalkSAT depends on structural parameter (we call odd Cheeger drift) of the dual of the formula hypergraph.
René Rodríguez-Aldama, Ajla Šehović, Enes Pasalic, Sadmir Kudin
An APN mapping $F:\mathbb{F}_{2^n}\to \mathbb{F}_{2^n}$ is a polynomial characterized by the non-vanishing property on 2-flats. In this work, we analyze notions that are closely related to this property. To understand which $k$-flats of $\mathbb{F}_{2^n}$ remain flats under $F$, we study the $k$-breaking. The function $x^{-1}$ has been studied in the past in this context---we extend this study to general mappings and characterize the 2-breaking of APN functions. Recently, two generalizations of the APN property have been introduced: $k$-strongly non-normality and $k$-th-order sum-freedom. Sum-freedom generalizes the non-vanishing property of APN functions to higher dimensional flats. We provide in-depth observations of the relations between the breaking property, strongly non-normality and sum-freedom. We introduce a fourth concept called $k$-strongly breaking, which implies the breaking property. We derive several structural results for both notions and give a characterization of a subclass of APN functions in terms of the 2-strongly breaking property. We propose a different perspective of the non-vanishing property via a natural character transformation, which is closely related to the sum-of-square indicator of the components of $F$. We derive a precise value for the total sum of the sum-of-square indicators of $F$. With this approach, we provide a simple answer to Open Problem 4 in IEEE Trans. Inf. Theory 52(9): 4160-4170, 2006. Moreover, it allows us to explore balancedness properties of polynomials, one of which characterizes component-wise APNness, for odd $n$, and provides a natural extension to any dimension. We show that Dillon's APN permutation and the Gold functions satisfy a related property, termed $k$-balanced, which is presented under our framework.
Let $P$ be a polytope defined by the system $A x \leq b$, where $A \in R^{m \times n}$, $b \in R^m$, and $\text{rank}(A) = n$. We give a short geometric proof of the following tight upper bound on the number of vertices of $P$: $$ n! \cdot \fracΔ{Δ_{\text{average}}} \cdot \text{vol}(B_2) \sim \frac{1}{\sqrt{πn}} \cdot \left(\frac{2 π}{e}\right)^{n/2} \cdot n^{n/2} \cdot \fracΔ{Δ_{\text{average}}}, $$ where $Δ$ is the maximum absolute value of $n \times n$ subdeterminants of $A$, and $Δ_{\text{average}}$ is the average absolute value of subdeterminants of $A$ corresponding to a triangulation of $P$'s normal fan. Assuming that $A$ is integer, such polyhedra are called $Δ$-modular polyhedra. Note that in the integer case, the bound can be simplified via the inequality $Δ_{\text{average}} \geq Δ_{\min} \geq 1$, where $Δ_{\min}$ is the minimum absolute value of subdeterminants of $A$ corresponding to feasible bases of $A x \leq b$. For this, we prove and use a symmetric variant of Macbeath's theorem. Additionally, we give a direct argument based on prior results in the field, showing that the graph diameter of $P$ is bounded by $O\bigl(n^3 \cdot \fracΔ{Δ_{\min}} \cdot \ln (n \fracΔ{Δ_{\min}}) \bigr)$. Thus, both characteristic of $P$ are linear in $Δ/Δ_{\min}$. From an algorithmic perspective, we demonstrate that: Given $A \in Q^{m \times n}$, $b \in Q^m$, and an initial feasible solution to $A x \leq b$, the convex hull of $P$ can be constructed in $O(n)^{n/2} \cdot m^2 \cdot \fracΔ{Δ_{\text{average}}}$ operations. For simple polyhedra, the dependence on $m$ reduces to linear; Given $A \in Z^{m \times n}$ and $b \in Q^m$, the number $|P \cap Z^n|$ can be computed in $O(n)^n \cdot \frac{Δ^4}{Δ_{\text{average}}}$ arithmetic operations.
Gonzalo Rodríguez-Pajares, Diego Ruano, Flavio Salizzoni
We give a combinatorial proof that the set of points associated to a self-dual code with no proportional columns is arithmetically Gorenstein if and only if the code is indecomposable. This answers a question asked by Tohăneanu. We do so by providing a combinatorial way to compute the dimension of the Schur square of a self-dual code through a zero-one symmetrization of a systematic generator matrix. Our approach also allows us to compute the Gorenstein defect. As a consequence, we obtain a combinatorial characterization of arithmetically Gorenstein self-associated sets of points over an algebraically closed field. We also show that, for any finite field, almost all self-dual codes of large length are indecomposable, and hence their associated sets of points are arithmetically Gorenstein.
Local-to-Global theorems are a cornerstone of many of recent developments in theoretical computer science from coding theory and PCPs all the way to analyzing mixing time of Markov chains. Building on trickledown theorems on simplicial complexes, several techniques have been established to obtain tighter local-to-global theorems to analyze the spectral gap of the Glauber dynamics, namely matrix trickledown, ${\mathcal C}$-Lorentzian polynomials and more recently Bochner type inequalities. In this paper, we adopt non-local local-to-global/Bochner type inequalities on "cones", corresponding to the local neighborhood of a given vertex, to show that the Glauber dynamics mixes in polynomial time to sample proper colorings of maximum degree $Δ$ graphs as long as $q\geq 1.7408Δ$.
How many fixed points does belief propagation have on a binary pairwise model? Over complex parameters, the admissible equations of a finite loopless multigraph form an algebraic family whose generic degree we determine: it is a graph state sum over edge sets without vertices of degree one, each vertex of positive degree contributing one less than its degree. We give two geometric realizations of this state sum. The first is the total Betti number of a universal edge-incidence ring constructed without parameters. Its full graded Betti table is the state sum, its Tor algebra is natural for noncontracting graph maps, the alternating numerator recovers the graph polynomial of Watanabe and Fukumizu, and an Artin reduction by a regular sequence realizes the one-variable specialization, corrected for tree components. The second lies in the actual family: at the strong-field boundary each state factor is a ramification index, and at sufficiently strong field the state charts exhaust the admissible complex fiber. For fixed nonzero real couplings, every Eulerian message-sign sector contains a real fixed point at every external field, and strong fields attain this lower bound: one admissible real fixed point per element of the binary cycle space and one positive fixed point. For graphs of maximum degree at most three, this minimum equals the number of minimal module generators of the edge-incidence ring. One state decomposition connects the complex degree, graded syzygies, and an attained minimum real count. The degree, selected ring and resolution statements, and the real counts are formalized in Lean 4.
We count Boolean functions represented as the sign of a real polynomial of bounded degree on the $n$-dimensional cube. We prove that the logarithm of their number is asymptotic to that of the classical hyperplane-arrangement upper bound, uniformly over all degrees, with relative error $O((\log n)^2/n)$. We also obtain sharper additive and relative error estimates in particular degree ranges, including an additive error $O(n^2)$ between the logarithms throughout a small linear degree range and improved bounds just below the central degree. The proof combines probabilistic estimates for Reed--Muller codes with a deletion argument for sets of evaluation vectors, an entropy count, and a sharp-threshold estimate for expected rank deficiency.
We study the \emph{Polynomial Avoidance Problem}: given a finite collection $F \subseteq Z[x_1,\dots,x_n]$ of nonzero polynomials, find a vector $x \in Z^n$ of minimum $\ell_p$-norm such that $f(x)\neq0$ for every $f \in F$. Each polynomial is given sparsely as a list of its nonzero terms. The affine special case is the \emph{Hyperplane Avoidance Problem}. This special case arises in Barvinok-type algorithms for counting integer points in polyhedra, when certain rational generating functions must be evaluated at a removable singularity. Our main results are as follows: 1) Let $D:=\sum_{f\in F} \text{deg} f$. Every instance admits a feasible vector $x$ satisfying \[ \|x\|_1 \leq \min\left\{D,\frac{D+n}{2}\right\}. \] This bound is tight. Such a vector can be found by a deterministic algorithm whose bit complexity is polynomial in $D$ and the sparse encoding length of the input. Our proof uses a new algorithmic variant of the \emph{Combinatorial Nullstellensatz}; 2) In the affine case, with $m$ input polynomials, such a vector can be constructed by a fast deterministic polynomial-time algorithm using $O(nm)$ arithmetic operations. For the homogeneous case, Barvinok's original approach provides only the weaker guarantee $\|x\|_1 = O((nm)^{n-1})$; 3) The problem is NP-hard for every $p\in R_{\geq 1}\cup\{\infty\}$, even for homogeneous linear forms; 4) As an application, we improve the complexity bound for counting integer points in general rational polytopes. We also improve the bound for polyhedra of bounded codimension; this includes the counting version of the \emph{Unbounded Subset-Sum} problem as a special case.
In 1984, Frank Harary introduced the first convexity games in graphs, all of them based on the geodesic convexity, which is the graph convexity related to shortest paths. In 2024, Araújo et al. obtained the first PSPACE-hardness proofs on some of these geodesic games and generalized them to any graph convexity. In this paper, we investigate convexity games on several known induced path convexities: the monophonic $\mathrm{m}$-convexity and the $\ell_k$-convexities, based on induced paths and on induced paths of size at most $k$. We prove that the hull games $\mathrm{CHG}_{\mathrm{m}}$, $\mathrm{CHG}_{\ell_k}$ and their misère variants are PSPACE-complete for every $k\ge2$ even in graphs with diameter at most 3. We also use the Sprague-Grundy Theory to obtain a polynomial time algorithm to decide the winner of the games $\mathrm{CHG}_{\mathrm{m}}$ and $\mathrm{CHG}_{\ell_k}$ for any $k\ge2$ in disjoint unions of paths and cycles. For $k\ge3$ odd, we prove that Alice (1st player) wins $\mathrm{CHG}_{\ell_k}$ in the path $P_n$ if and only if $n$ is odd and she wins in the cycle $C_n$ if and only if $n=3$ or $n=α\cdot (k+1)-1$ with $α\ge2$. For $k\ge2$ even, the only periodic nimber sequences of $\mathrm{CHG}_{\ell_k}$ obtained through extensive computational testing occurred for $k=2^h-4$ with $h\ge3$, e.g, $k\in\{4,12,28,60,\ldots\}$. In this case ($k=2^h-4$ with $h\ge3$), we prove that the nimber sequences of $\mathrm{CHG}_{\ell_k}$ in $P_n$ and in $C_n$ are periodic and Alice loses (resp. wins) in $P_n$ (resp. $C_n$) with $n>k$ only when $n=3k+4$ (resp. $n\in\{2k+1,5k+3\}$). Finally, we show that, for $k=2$, the game $\mathrm{CHG}_{\ell_2}$ in paths $P_n$ is closely related to the classical game \emph{Couples-are-Forever} of J. H. Conway: it is still an open problem if the nimber sequence is periodic or not and Alice loses only for 12 values of $n$ up to $150$ million.
Saul D. Freedman, Melissa Lee, Dragan Marušič, Tomaž Pisanski
Given a finite graph $G$, we characterize the automorphisms and automorphism groups of $G$ that can be realized as geometric symmetries and symmetry groups of straight line drawings of $G$ in the plane. In doing so, we correct a previous version of this characterization from the literature. As an application, we show that the graphs recently constructed by the first two authors to disprove the Polycirculant Conjecture, which have orders $16{,}464^k$ for each positive integer $k$, are the first known vertex-transitive graphs with no non-trivial realizable geometric symmetries. We also construct infinite families of graphs that exhibit all possible kinds of geometric symmetry.
A tree cover of an $n$-point metric space is a collection of $k$ trees such that every pairwise distance is approximately preserved in at least one tree. For fixed $k$, the best known distortion upper bound is $\tilde{O}(n^{1/k})$ and the best known lower bound is $Ω_k(n^{1/2^{k-1}})$. In this paper, we prove a lower bound of $Ω_k(n^{1/(k-1)^2})$ for all $k\ge 3$, significantly narrowing the gap towards the upper bound. Our approach makes use of a novel algebraic topological notion that we call the \emph{agreement index}: the equivariant topological index of a simplicial complex encoding pairwise agreement among tree labels. This notion helps characterize the power and limit of algebraic and topological methods for analyzing tree covers, and we believe it will also find applications to other problems.
If $G$ and $H$ are two cubic graphs, then an $H$-coloring of $G$ is a mapping $f: E(G)\rightarrow E(H)$, such that for every $v\in V(G)$ there is a vertex $x\in V(H)$, such that $f(\partial_G(v))=\partial_H(x)$. If $G$ admits an $H$-coloring then it is common to write $H\prec G$. In this paper, for a fixed connected cubic graph $H$, we consider a decision problem, which we call $H$-coloring. In this problem, on the input one is given a connected cubic graph $G$. The goal is to decide whether $H\prec G$. A conjecture by Hakobyan and the author states that $H$-coloring is polynomial time solvable, if $H$ does not admit a $P_{12}$-coloring. Here $P_{12}$ is the unique cubic graph obtained from the Petersen graph by expanding one of its vertices into a triangle. In this paper, we disprove this conjecture under $P\neq NP$, by showing that $S_6$-coloring is NP-complete. Here $S_6$ is the unique cubic graph on six vertices having exactly one bridge. Our second result is an algorithm that solves $H$-coloring problem in time $O^*(c^n)$, where $H$ is a simple connected cubic graph, $n$ is the number of vertices of the input cubic graph $G$ and $c=\sqrt{8}\approx 2.8284$.
Nicholas J. Cooper, François G. Meyer, Michael L. Roberts, Carlos Zapata-Carratalá, Lijun Chen, Danna Gurari
We introduce a unified theoretical framework for the rigorous analysis and systematic construction of deep neural networks (DNNs). This framework addresses a gap in existing theory by explicitly modeling the structure of tensor operations -- lower level information that is often abstracted. Our framework enables two novel objectives: (1) analysis of the evolution of architectural complexity over deep learning history, and (2) automatic construction of novel architectures based on new types of tensor operations. Our study of DNNs introduced over the past 40 years reveals a connection between groundbreaking architectures and increases in different types of architectural complexity. Moreover, we identify several large classes of higher complexity architectures that have not yet been explored. We then collect a dataset of 3,000+ higher complexity architectures, which we publicly release at: https://github.com/combinatoriallabs/ArchitecturalComplexity.
An ordered graph is a graph with a total order over its vertices. A linear layout of an ordered graph is a partition of the edges into sets of either non-crossing edges, called stacks, or non-nesting edges, called queues. The stack (queue) number of an ordered graph is the minimum number of required stacks (queues). Mixed linear layouts combine these layouts by allowing each set of edges to form either a stack or a queue. The minimum number of stacks plus queues is called the mixed page number. It is well known that ordered graphs with small stack number are characterized, up to a function, by the absence of large twists (that is, pairwise crossing edges). Similarly, ordered graphs with small queue number are characterized by the absence of large rainbows (that is, pairwise nesting edges). However, no such characterization via forbidden patterns is known for mixed linear layouts. We address this gap by introducing patterns similar to twists and rainbows, which we call thick patterns; such patterns allow a characterization, again up to a function, of mixed linear layouts of bounded-degree graphs. That is, we show that a family of ordered graphs with bounded maximum degree has bounded mixed page number if and only if the size of the largest thick pattern is bounded. In addition, we investigate an exact characterization of ordered graphs whose mixed page number equals a fixed integer $ k $ via a finite set of forbidden patterns. We show that for every $ k \ge 2 $, there is no such characterization, which supports the nature of our first result.
Let $(T, \circ)$ be a loop of order $k \geq 3$ and $D_S^*$ the loopless central digraph obtained from the associated central digraph $D_S$ by removing the $k$ self-loops at the diagonal vertices. We construct an explicit arc-labeling $f$ of $D_S^*$ ordering all arcs lexicographically by $(\text{tail}, \text{head})$ and assigning labels $1, \ldots, m$ in that order and prove that $f$ is an antimagic labeling. Consequently, $D_S$ is antimagic for every loop of order $k \geq 3$, a property that depends only on $k$ and is invariant across all non-isomorphic loops of the same order.
Let $d_{K_{s,t}}(n)$ be the largest fraction of all graphs on $[n]$ in a family containing no two distinct members whose symmetric difference is a copy of $K_{s,t}$. We determine this maximum density for every fixed pair of positive integers $s,t$. If both part sizes are odd, it is known that the density is exactly $1/2$. Otherwise, let $2h$ be the largest even part size. We prove $$ d_{K_{s,t}}(n)=Θ_{s,t}(n^{-h}). $$ For $h>1$, this improves the general $O_{s,t}(1/n)$ upper bound obtained from Alon's two-copy gluing theorem~\cite{Alon2024}. The sharp bounds for even stars were already proved by Alon and are recovered as special cases. The upper bound uses the Frankl--Füredi forbidden-intersection theorem. For the lower bound, we construct binary linear codes using BCH codes. When exactly one part size is even, we assign to each edge the sum of the binary vectors assigned to its endpoints. When both part sizes are even, we identify these vectors with elements of a finite field and assign to each edge the product of its endpoint elements. In both cases, every copy of $K_{s,t}$ has a nonzero syndrome.
The running-time distribution of a randomized algorithm is computed under independent randomness, but independence is rarely what a source guarantees. We study sources $ν$ satisfying $ν[w]\le DP[w]^s$ for every finite prefix $w$ of the random tape, where $P$ is an independent reference law, $0<s\le1$, and $D\ge1$; individual choices may be predictable, even determined by the past. The worst-case probability of any closed tape event $E$ is exactly $\min\{1,DC_s(E)\}$, where $C_s(E)$ is the infimum cost of covering $E$ by prefixes of varying depth, a tree max-flow/min-cut identity that makes worst-case survival probabilities exact and optimizable over the algorithm's choices. Three applications follow. For $n$ tasks on one machine, each retried until the next random symbol lies in its success set, every schedule has the same makespan distribution under independent randomness, yet under our constraint two fixed orders miss a linear deadline with worst-case probabilities $1$ and $\exp[-Θ(n)]$; a static priority rule is optimal at every deadline. For the standard local walk on a $k$-CNF with an assignment that satisfies a majority of each clause, we determine the sharp source threshold $s_*$. Below $s_*$, on some instances a single admissible source forces nontermination however clauses are selected; at $s_*$, with fixed $D$, the expected number of flips is at most cubic in the initial Hamming distance, against linear under independence, with matching worst-case instances; and any fixed margin above $s_*$ restores linear dependence on the distance. Third, verified backtracking for noisy predecessor search remains correct under a sufficiently strong bound on the comparison errors. In every setting the algorithm may also restart, which supplies no fresh randomness; restarting restores exponential tails exactly when the nontermination event has a prefix cover of cost below one.
We give a deterministic polynomial-time algorithm for the Bilu--Linial signing problem on bipartite graphs. For every finite simple bipartite graph of maximum degree at most an integer $Δ\ge3$, the algorithm assigns signs to its edges so that the signed adjacency matrix has operator norm strictly less than $2\sqrt{Δ-1}$. Our algorithm builds on the randomized recursive repair framework of Jadbabaie, Saberi, and Sra~\cite{JSS26}, with deterministic rules for sign selection and vertex deletion.
Benade et al. (EC 2024) introduced a semi-random model for non-additive fair division in which each agent starts with an arbitrary base valuation over bundles of items and independently permutes the item names uniformly at random. While they proved that envy-free allocations exist with high probability as the number of items grows for $n = 2$ agents, they left open whether the same holds for $n > 2$ agents. We resolve this open question in the affirmative: for any fixed number of agents $n$ and any number of items $m$ divisible by $n$, an envy-free allocation exists with probability $1 - o(1)$ as $m \to \infty$. Our proof combines the second moment method with a connection to the (small-set) expansion of generalized Johnson graphs, which may be of independent interest.
We present improved lower bounds for nine classical Ramsey numbers: $\mathbf{R}(3, 13)$ is increased from $60$ to $61$, $\mathbf{R}(3, 18)$ from $99$ to $100$, $\mathbf{R}(4, 13)$ from $138$ to $139$, $\mathbf{R}(4, 14)$ from $147$ to $148$, $\mathbf{R}(4, 15)$ from $158$ to $159$, $\mathbf{R}(4, 16)$ from $170$ to $174$, $\mathbf{R}(4, 18)$ from $205$ to $209$, $\mathbf{R}(4, 19)$ from $213$ to $219$, and $\mathbf{R}(4, 20)$ from $234$ to $237$. These results were achieved using AlphaEvolve, an LLM-based code mutation agent. Beyond these new results, we successfully recovered lower bounds for all Ramsey numbers known to be exact, and matched the best known lower bounds across many other cases. These include bounds for which previous work does not detail the algorithms used. Virtually all known Ramsey lower bounds are derived computationally, with bespoke search algorithms each delivering a handful of results. AlphaEvolve is a single meta-algorithm yielding search algorithms for all of our results.
Can AI based methods help us make advances in complexity theory? We provide evidence towards answering this in the affirmative, using AlphaEvolve (an LLM code mutation agent) to obtain new results in three settings: a) We improve a recent result of Kunisky and Yu to obtain near-optimal upper and (conditional) lower bounds on certification algorithms for MAX-CUT and MAX-Independent Set on random 3- and 4-regular graphs. Our improved lower bounds are obtained by constructing nearly extremal Ramanujan graphs on as many as $163$ vertices, and our upper bounds are obtained via analytical arguments. b) We obtain new inapproximability results for MAX-4-CUT and MAX-3-CUT, proving that it is NP-hard to approximate them within factors of $0.987$ and $0.9649$ respectively, using AlphaEvolve to discover new gadget reductions. Our MAX-4-CUT result improves upon the SOTA of $0.9883$, and our MAX-3-CUT result improves on the current best gadget-based inapproximability result of $0.9853$, but falls short of the SOTA of $16/17$ that relies on a custom PCP (rather than a reduction from ``standard'' Håstad-style PCPs). c) Inapproximability for the metric Traveling Salesman Problem (TSP): We show that it is NP-hard to approximate the minimum cost tour within a factor of $111/110$ using AlphaEvolve to discover a new gadget, thus improving the SOTA of $117/116$. Along the way, we provide new modular soundness and completeness arguments that can be of independent interest. A key technical challenge we faced: verifying a candidate construction produced by AlphaEvolve is costly (sometimes requiring time exponential in the size of the construction). We used AlphaEvolve itself to evolve the verification procedure to be faster (sometimes by $10,000\times$ for our gadgets). Our results suggest that gadget based proofs would benefit from a pass through AI-based tools to obtain stronger results.
Tony Feng, Trieu Trinh, Garrett Bingham, Jiwon Kang, Shengtong Zhang, Sang-hyun Kim, Kevin Barreto, Carl Schildkraut 等 24 位作者
We present a case study in semi-autonomous mathematics discovery, using Gemini to systematically evaluate 700 conjectures labeled 'Open' in Bloom's Erdős Problems database. We employ a hybrid methodology: AI-driven natural language verification to narrow the search space, followed by human expert evaluation to gauge correctness and novelty. We address 13 problems that were marked 'Open' in the database: 5 through seemingly novel autonomous solutions, and 8 through identification of previous solutions in the existing literature. Our findings suggest that the 'Open' status of the problems was through obscurity rather than difficulty. We also identify and discuss issues arising in applying AI to math conjectures at scale, highlighting the difficulty of literature identification and the risk of ''subconscious plagiarism'' by AI. We reflect on the takeaways from AI-assisted efforts on the Erdős Problems.