We resolve a $1,000 Erdős prize problem, complete with formal verification generated by a large language model. In over a dozen papers, beginning in 1976 and spanning two decades, Paul Erdős repeatedly posed one of his "favorite" conjectures: every finite Sidon set can be extended to a finite perfect difference set. We establish that {1, 2, 4, 8, 13} is a counterexample to this conjecture. During the preparation of this paper, we found that although this problem was presumed to be open for half a century, Marshall Hall, Jr. published a different counterexample three decades before Erdős first posed the problem. With a healthy skepticism of this apparent oversight, and out of an abundance of caution, we used ChatGPT to vibe prove both Hall's and our counterexamples in Lean.

Things can go spectacularly wrong when clustering timeseries data that has been preprocessed with a sliding window. We highlight three surprising failures that emerge depending on how the window size compares with the timeseries length. In addition to computational examples, we present theoretical explanations for each of these failure modes.

What fraction of points in a planar Poisson process are not the nearest neighbor of any other point? That is, what is the probability that a given particle is not the nearest neighbor of any other particle in a classical two-dimensional ideal gas model? In 1987, Tao and Wu presented an award-winning tour de force of integration to solve this problem, but in the time since, errors were discovered in their solution, and the technical difficulty precluded researchers from correcting them. In this paper, we rectify this situation by providing a complete solution to the original problem. First, we prove a correct version of Tao and Wu's intended change of variables, and in doing so, we identify an important constraint that was missing in one of the integration variables. Next, we use ideas from Fourier–Motzkin elimination to prove bounds of integration that correctly incorporate this missing constraint. The result is a weighted sum of two double integrals, four quadruple integrals, four sextuple integrals, and two octuple integrals, and we apply a variety of techniques to numerically compute each of these integrals. Considering the error-riddled history of this problem, we conclude by discussing the various ways we verified different portions of our solution.

Motivated by a popular code golf challenge, we review some key ideas from information theory and discuss how to efficiently compress a streaming file with an acceptable error rate.

In this paper, we study approximate Hadamard matrices, that is, well-conditioned n× n matrices with all entries in {±1}. We show that the smallest-possible condition number goes to 1 as n→∞, and we identify some explicit infinite families of approximate Hadamard matrices.

We determine the largest subset A⊆{1,…,n} such that for all a,b∈ A, the product ab is not squarefree. Specifically, the maximum size is achieved by the complement of the odd squarefree numbers. This resolves a problem of Paul Erdős and András Sárközy from 1992.

For a uniformly distributed population, we show that with high probability, any majority-minority voting district containing a fraction of the population necessarily exhibits a tiny Polsby-Popper score.

We improve the best known upper bound on the number of edges in a unit-distance graph on n vertices for each n∈{15,…,30}. When n≤ 21, our bounds match the best known lower bounds, and we fully enumerate the densest unit-distance graphs in these cases. On the combinatorial side, our principle technique is to more efficiently generate ℱ-free graphs for a set of forbidden subgraphs ℱ. On the algebraic side, we are able to determine programmatically whether many graphs are unit-distance, using a custom embedder that is more efficient in practice than tools such as cylindrical algebraic decomposition.

We correct errors that appear throughout "The vicious neighbour problem" by Tao and Wu. We seek to solve the following problem. Suppose N nerds are distributed uniformly at random in a square region. At 3:14pm, every nerd simultaneously snipes their nearest neighbor. What is the expected proportion P_N of nerds who are left unscathed in the limit as N→∞?

Bizarrely shaped voting districts are frequently lambasted as likely instances of gerrymandering. In order to systematically identify such instances, researchers have devised several tests for so-called geographic compactness (i.e. shape niceness). We demonstrate that under certain conditions, a party can gerrymander a competitive state into geographically compact districts to win an average of over 70% of the districts. Our results suggest that geometric features alone may fail to adequately combat partisan gerrymandering.

The U.S. Supreme Court is currently deliberating over whether a proposed mathematical formula should be used to detect unconstitutional partisan gerrymandering. We show that in some cases, this formula will only flag bizarrely shaped districts as potentially constitutional.

In many areas of imaging science, it is difficult to measure the phase of linear measurements. As such, one often wishes to reconstruct a signal from intensity measurements, that is, perform phase retrieval. In this paper, we provide a novel measurement design which is inspired by interferometry and exploits certain properties of expander graphs. We also give an efficient phase retrieval procedure, and use recent results in spectral graph theory to produce a stable performance guarantee which rivals the guarantee for PhaseLift in [Candes et al. 2011]. We use numerical simulations to illustrate the performance of our phase retrieval procedure, and we compare reconstruction error and runtime with a common alternating-projections-type procedure.

Finite frame theory has a number of real-world applications. In applications like sparse signal processing, data transmission with robustness to erasures, and reconstruction without phase, there is a pressing need for deterministic constructions of frames with the following property: every size-M subcollection of the M-dimensional frame elements is a spanning set. Such frames are called full spark frames, and this paper provides new constructions using the discrete Fourier transform. Later, we prove that full spark Parseval frames are dense in the entire set of Parseval frames, meaning full spark frames are abundant, even if one imposes an additional tightness constraint. Finally, we prove that testing whether a given matrix is full spark is hard for NP under randomized polynomial-time reductions, indicating that deterministic full spark constructions are particularly significant because they guarantee a property which is otherwise difficult to check.

Link-based data structures, such as linked lists and binary search trees, have many well-known rearrangement steps allowing for efficient implementations of insertion, deletion, and other operations. We describe a rearrangement primitive designed for link-based, heap-ordered priority queues in the comparison model, such as those similar to Fibonacci heaps or binomial heaps. In its most basic form, the primitive rearranges a collection of heap-ordered perfect binary trees. Doing so offers a data structure control on the number of trees involved in such a collection, in particular keeping this number logarithmic in the number of elements. The rearrangement step is free from an amortized complexity standpoint (using an appropriate potential function).

In the course of proving the strong perfect graph theorem, Chudnovsky, Robertson, Seymour, and Thomas showed that every perfect graph either belongs to one of five basic classes or admits one of several decompositions. Four of the basic classes are closed under the operation of taking induced subgraphs (and have known forbidden subgraph characterizations), while the fifth one, consisting of double-split graphs, is not. A graph is doubled if it is an induced subgraph of a double-split graph. We find the forbidden induced subgraph characterization of doubled graphs; it contains 44 graphs.

We prove that the probability that one of the players has a winning strategy in the War-like card game "Peer Pressure" approaches zero as the number of cards dealt approaches infinity, even if the cards are dealt in a slightly-biased manner.

The results of Strassen~\cite{strassen-tensor} and Raz~\cite{raz} show that good enough tensor rank lower bounds have implications for algebraic circuit/formula lower bounds. We explore tensor rank lower and upper bounds, focusing on explicit tensors. For odd d, we construct field-independent explicit 0/1 tensors T:[n]^d\to\mathbb{F} with rank at least 2n^{\lfloor d/2\rfloor}+n-\Theta(d\lg n). This improves the lower-order terms in known lower bounds for any odd d\ge 3. We also explore a generalization of permutation matrices, which we denote permutation tensors. We show, by applying known counting lower bounds, that there exist order-3 permutation tensors with super-linear rank as well as order-$d$ permutation tensors with high rank. We also explore a natural class of permutation tensors, which we call group tensors. For any group G, we define the group tensor T_G^d:G^d\to\mathbb{F}, by T_G^d(g_1,\ldots,g_d)=1$ iff $g_1\cdots g_d=1_G. We give two upper bounds for the rank of these tensors. The first uses representation theory and works over ``large'' fields $\mathbb{F}, showing (among other things) that \rank_\mathbb{F}(T_G^d)\le |G|^{d/2}. In the case that d=3, we are able to show that \rank_\mathbb{F}(T_G^3)\le O(|G|^{\omega/2})\le O(|G|^{1.19}), where $\omega$ is the exponent of matrix multiplication. The next upper bound uses interpolation and only works for abelian G, showing that over any field \mathbb{F}$ that $\rank_\mathbb{F}(T_G^d)\le O(|G|^{1+\lg d}\lg^{d-1}|G|). In either case, this shows that many permutation tensors have far from maximal rank, which is very different from the matrix case and thus eliminates many natural candidates for high tensor rank. We also explore monotone tensor rank. We give explicit 0/1 tensors T:[n]^d\to\mathbb{F} that have tensor rank at most $dn$ but have monotone tensor rank exactly n^{d-1}. This is a nearly optimal separation.

A linear equation L is called k-regular if every k-coloring of the positive integers contains a monochromatic solution to L. Richard Rado conjectured that for every positive integer k, there exists a linear equation that is (k - 1)-regular but not k-regular. We prove this conjecture by showing that the equation Sigma(k-1)(i=1) 2(i)/2(i)-1x(i) = (-1+Sigma(k-1)(i=1) 2(i)/2(i)-1)x(0) has this property.This conjecture is part of problem E14 in Richard K. Guy's book "Unsolved Problems in Number Theory", where it is attributed to Rado's 1933 thesis, "Studien zur Kombinatorik". (C) 2009 Elsevier Inc. All rights reserved.

We show that inverse problems with a truncated quadratic regularization are NP-hard in general to solve or even approximate up to an additive error. This stands in contrast to the case corresponding to a finite-dimensional approximation to the Mumford-Shah functional, where the operator involved is the identity and for which polynomial-time solutions are known. Consequently, we confirm the infeasibility of any natural extension of the Mumford-Shah functional to general inverse problems. A connection between truncated quadratic minimization and sparsity-constrained minimization is also discussed.

Recent research Boris Alexeev I am currently working on an induced subgraph characterization of double-split graphs. (Recall that double-split graphs are one of the ”basic classes” in the proof of the strong perfect graph theorem.) On Tuza’s Conjecture Jessica McDOnald Tuza conjectured that for every graph G, the maximum size of a set of edge disjoint triangles is at most twice the minimum size of a set of edges meeting every triangle. Krivelevich proved that this result is true if we replace the set of edges with a fractional set and asked if any further improvement on this bound might be possible. We achieve an essentially best possible bound relating these two parameters. We also consider a related problem for weighted graphs. This is joint work with Guillaume Chapuy, Matt DeVos, Bojan Mohar, and Diego Scheide. Growth in Cubes of Graphs Matt DeVos Let A be a generating set of a finite multiplicative group. A number of important problems in additive number theory and group theory concern the increase in size from |A| to |Ak|. This can be stated in terms of graphs by comparing the number of edges in the Cayley Graph generated by A compared with the number in the Cayley Graph generated by Ak. Motivated by this connection, Hegarty considered this problem in more general graphs. He proved that e(G3) > (1 + c)e(G) for every connected regular graph of diameter at least 3 and asked what the best possible constant is in this theorem. We show that c = 3/4 is best possible. Joint with Stephan Thomasse. Recent research Zdenek Dvorak Recently, I was considering various problems related to 4-color theorem, as well as alternative approaches to its proof (with only a little progress, unfortunately). Examples: – Grotzsch conjecture regarding the 3-edge-colorability of subcubic planar graphs – characterization of 5-critical graphs with crossing number 1 – characterization of non-4-colorable Eulerian triangulations of bounded genus – for each cubic planar graph G with a fixed number of half-edges incident with the outer face and each precoloring psi of these half-edges, we can consider the number c(G,psi) of the 3-edge-colorings of G that extend psi. Can we characterize the possible vectors c(G,.)? Using Kochol’s technique and blockreducibility arguments, we can obtain a linear program constraining these vectors; can this characterization be strenghtened? A new proof of the graph removal lemma Jacob Fox/Massachusetts Institute of Technology