Papers

We prove that the Füredi-Hajnal limit and the Stanley-Wilf limit of a uniformly random permutation matrix of order $k$ are at most $\exp\bigl(O(\sqrt{k}(\log k)^{5/2})\bigr)$ with probability tending to one as $k\to\infty$. This improves the bound $\exp\bigl(O(k^{2/3}(\log k)^{7/3}/(\log\log k)^{1/3})\bigr)$ of Cibulka and Kynčl. Together with the lower bound due to Fox, these bounds show that the logarithms of both limits are $k^{1/2+o(1)}$ for almost all permutations.

We prove that, for every $\eps>0$, any $n$-vertex graph that needs at least $\eps n^2$ edge changes to become a Turán graph has positive discrepancy at least $c_\eps n^{5/4}$. Consequently, every such regular graph has second eigenvalue at least $c'_\eps n^{1/4}$. These results prove two conjectures of Räty, Sudakov and Tomon.

Problem 73 on Green's list of open problems asks whether a smooth submanifold of codimension two in $\mathbb{R}^n$ must meet some affine two-plane in five points when $n$ is large. In this paper, we answer this question for every $n \geq 6$. Every nonempty relatively open subset of such a submanifold contains the vertices of an affinely regular pentagon, and the bound $n \geq 6$ is sharp for affinely regular pentagons. More generally, for every prime $p \geq 5$, the graph of a continuous map from a ball in $\mathbb{R}^{p-1}$ to $\mathbb{R}^2$ contains an affinely regular $p$-gon. A second argument gives six coplanar points when $n \geq 7$. Both proofs find the points by making the low Fourier modes of the graph map vanish along a circle.

For some integer $\ell\geq 2$, consider $m=2\ell-1$ voters, each of which has a preference ranking over $n$ alternatives. The Smith set is the smallest nonempty set of alternatives each of which defeats every alternative outside the set in a pairwise majority comparison. Under the standard benchmark impartial culture where each voter uniformly and independently chooses a random preference ranking over all alternatives, we study the asymptotic probability that the Smith set has cardinality $s$ for fixed $\ell$ and $n\to \infty$. First, we prove that the probability that the Smith set has constant cardinality $s$ goes to zero at a rate of $\Theta_{\ell,s} (n^{-s(\ell-1)/\ell})$. Next, if $\min\{s,n-s\}\to\infty$, we show that this probability decays superpolynomially. Further, when $s$ and $n-s$ are both $\Theta(n)$, we prove that the probability decays exponentially and determine the exact exponential base. Finally, we prove that the probability that the Smith set contains all alternatives approaches one at a rate of $\Theta_\ell(n^{-(\ell-1)/\ell})$. These theorems resolve the conjectures in [Liu et al, Ann Stat 2026] under impartial culture.

A reduced word of the longest permutation $w_0\in S_m$, taken up to commutations, encodes a wiring diagram of $m$ pseudolines, and a braid move $\sigma_i\sigma_{i+1}\sigma_i\leftrightarrow\sigma_{i+1}\sigma_i\sigma_{i+1}$ flips one of its triangles. We prove that deciding whether two wiring diagrams are within a given braid-move distance is NP-complete. Equivalently, deciding flip distance is NP-complete in the higher Bruhat order $B(m,2)$, for triangle flips of marked arrangements of pseudolines, and for rhombic tilings of a $2m$-gon; this answers an open question on flip graphs of pseudoline arrangements (SODA 2024). The reduction is from Vertex Cover: for every graph $G$ we construct two wiring diagrams at braid-move distance $|D|+2\,\mathrm{VC}(G)$, where $\mathrm{VC}(G)$ is the vertex cover number of $G$ and the Hamming distance $|D|$ is the number of triples of wires whose triangles are oriented differently. The construction glues copies of a seven-wire gadget whose braid-move distance exceeds its Hamming distance by $2$; flipping either of two disjoint triangles in both diagrams removes the excess. Adding a line at infinity transfers the construction to rank-$3$ uniform oriented matroids, where two oriented matroids are adjacent in the mutation graph if their chirotopes differ in one basis; so deciding mutation distance is NP-complete in rank $3$. An iterate of Rambau's expansion of signotopes amplifies the excess and gives NP-hardness in every fixed rank $r\ge 4$. By duality we obtain the corresponding results for every fixed corank at least $3$. The same instances show that flip distance in the higher Bruhat orders $B(n,k)$ is NP-complete for every fixed $k\ge 2$.

In this paper, we provide a novel approach to limiting Heesch numbers of a specific family of tiles. In particular, we show how a certain family of three–dimensional prisms has a finite Heesch number. In so doing, we initiate a structural study of ratio $q$ tilings by two prototiles, in which, loosely speaking, one tile appears $q$ times more often than the other. The method works as follows: we take planar intersections with coronas around the prism in two perpendicular directions and compare the frequencies of the resulting prototiles in those patches. If every tiling by these prototiles has limiting ratio at least $q >1$, then sufficiently large coronas of the prism would require incompatible tile counts in the two directions. We also provide some computational results and exhibit $6$ infinite families of notched prisms that have finite Heesch numbers greater or equal to $2$. Finally, we provide some negative computational results about the non–existence of certain prisms with given Heesch numbers.

In 2007, Fomin and Zelevinsky introduced the bipartite belt, a sequence of bipartite cluster mutations whose exchange relations form a discrete dynamical system. For each Dynkin diagram, this system is periodic, as a special case of Zamolodchikov periodicity. In the associated tropical dynamics, every mutation along the belt can be naturally colored red or blue, according to which term attains the maximum in the tropical exchange relation. This gives a red-blue statistic on tropical orbits. We prove that this statistic is homomesic, i.e. that the average numbers of red and blue mutations are independent of the orbit. Moreover, the number of red mutations is given by the number of roots in the associated root system. This resolves a conjecture of the first author. In the process we introduce the Zamolodchikov fan, which is closely related to the positive tropical Grassmannian, and may be of independent interest.

We show that $\text{gr}(\text{Av}(1324))\leq 13.167248$. This is achieved by combining the approach of Bevan, Brignall, Elvey Price and Pantone using the `domino', with ideas from earlier upper bounds by Bóna using pairs of decorated words with additional restrictions. More specifically, we replace the arbitrary interleavings of Bevan et. al. with interleavings more like those of Bóna. Our method uses shuffle automata to handle the interleavings, and this method has the potential to provide better upper bounds than the one established here. To estimate how many shuffles are possible, we use two key statistics on dominoes: the number of internal points (that is, points that are neither left-to-right minima not right-to-left maxima), and the number of runs of internal points.

We study permutations $\pi \in S_n$ that avoid $123$ while satisfying the adjacency constraint $\lvert \pi_{i+1}-\pi_i\rvert \le m$ for some $m\in\mathbb{Z}^{+}$. We first show that every admissible permutation is localized within vertical distance $m$ of the reverse diagonal, revealing a strong global restriction imposed by the interaction between pattern avoidance and bounded adjacency. For every fixed $m$, we then construct an exact finite-state description of the family, with precisely $2^{m+1}-m-1$ realizable states. This yields an explicit transfer-matrix enumeration and, in particular, a rational generating function for every fixed $m$; consequently, the enumeration sequence satisfies an eventual constant-coefficient linear recurrence. We further identify the exponential growth rate with the spectral radius $\rho_m$ of the corresponding transfer matrix and show that $\rho_m<4$, is nondecreasing in $m$, and converges to the Catalan growth constant $4$ as $m\to\infty$. Finally, we show that the bounded-adjacency constraint breaks the ordinary length-three Wilf equivalence, and we develop algorithmic and graph-theoretic interpretations of the family, including a correspondence with constrained Hamiltonian paths in powers of the path graph.

We introduce and analyze a new model of graph burning, in which two competing fires (coloured yellow and green) ignite vertices of a given signed graph and propagate along its edges. The boolean sign of an edge determines whether a fire spreading along the edge changes colour or not. In each step, a player ignites a new vertex in a colour of their choice, while previously activated fires continue to spread. Given a signed graph $\Gamma$, the objective is to burn the maximum possible number of vertices in a single colour; the optimal achievable value is called the plurality number of $\Gamma$. We express the plurality number through Hamming distances to a binary code associated with $\Gamma$. In particular, the minimum plurality number over the switching class of $\Gamma$ equals the number of vertices minus the covering radius of this code. Under certain conditions on the signature, we determine exact values of the plurality number of signed paths. By contrast, we prove hardness results for multiple variants of the problem of determining or approximating the plurality number.

A parking function is said to have $\ell$-limited preferences if there are at most $\ell$ copies of each element in the parking function. In this paper, we define a notion of pattern avoidance for parking functions and enumerate those $2$-limited parking functions that avoid certain patterns.

In earlier work, we proved that the length polyhedron of an interval order has a unique Hilbert basis and that every element of this basis is a binary vector. The supports of these binary vectors form a set system whose intersection graph is a cocomparability graph. This leads naturally to the question of which graphs can arise as intersection graphs of such Hilbert-basis supports. We introduce Hilbert fragments, defined as induced subgraphs of these intersection graphs. Thus Hilbert fragments form a hereditary class of cocomparability graphs recording the intersection patterns among the supports of Hilbert-basis elements of interval-order length polyhedra. We characterize Hilbert fragments by an axiomatically defined vertex-labeling, called a track-labeling, and give applications illustrating how this characterization can reveal an underlying one-dimensional structure even when the Hilbert fragments themselves are not interval graphs.

We construct explicit non-malleable affine extractors for every constant entropy rate, with linear output length and exponentially small error, against any fixed number of affine tamperings without fixed points. For every fixed $0<\eta<1$ and $t$, we also obtain entropy threshold $C_{\eta,t}n/\log n$, output length $\lfloor n^{1-\eta}\rfloor$, and error $2^{-n^{1-\eta}}$ against $t$ tamperings. Our extractors, as well as the directional affine extractors of Li and Zhong (CCC 2024), yield explicit Boolean functions with correlation $2^{-\Omega(n)}$ against weakly read-once linear branching programs of size $2^{\Omega(n)}$. For non-oblivious decision trees, we prove linear depth lower bounds for queries of each fixed degree $r\ge2$. Applying Li's sumset extractor (FOCS 2023) gives depth $\Omega_\delta((n/\ell)\log\ell)$ for growing locality $\ell\le n^{1-\delta}$, where $0<\delta<1$ is fixed. In the same range, directional affine extractors give correlation $2^{-\Omega(n/\sqrt\ell)}$ against local trees of depth $c(n/\ell)\log\ell/\log\log\ell$, for a sufficiently small constant $c>0$. Our extractors derandomize the lossless lifting of Efremenko and Itsykson (STOC 2026). For every fixed $0<\xi<1$, this gives explicit polynomial-size unsatisfiable CNFs on $N$ variables whose $\mathrm{Res}(\oplus)$ refutations of resolution depth at most $N$ require size at least $2^{(1-\xi)N}$. Separately, parity substitutions give polynomial-size CNFs on $N$ variables with polynomial-size ordinary-resolution proofs for which every $\mathrm{Res}(\oplus)$ refutation of size $S$ and depth $d$ satisfies $d\log(2S)=\Omega(N^2)$. This removes the $\log^2 N$ loss in the tradeoff of Itsykson, Podolskii, and Shekhovtsov (CCC 2026).

We construct near-spanning balanced antidirected trees that fail to embed in highly symmetric tournaments despite the conjectured semidegree condition being satisfied. More precisely, for every sufficiently large odd integer $n$, we exhibit a regular vertex-transitive tournament on $n$ vertices and a balanced antidirected caterpillar on $n-1$ vertices with maximum degree at most $(1+o(1))n/\log_2 n$ that is not contained in the tournament. The host satisfies the strict $k/2$ threshold in both the semidegree and pseudo-semidegree settings, where $k$ is the number of arcs of the target tree. Consequently, this gives counterexamples to the uniform sublinear-degree interpretations of Conjectures~6.8 and~7.6 in Stein's survey (2024). The order $n-1$ of the target is best possible under the strict semidegree hypothesis. We also give a six-vertex balanced antidirected double-star missing from the seven-vertex Paley tournament, and show that seven is the smallest host order for a pseudo-semidegree counterexample with this fixed double-star.

Using the equivalence of sphere order representations of posets with Minkowski space embeddings, we show that the Boolean lattice $\mathcal{B}_6$ is not a sphere order. This is the first explicit finite example of a poset that is not a sphere order.

The union-closed sets conjecture asks whether every finite union-closed family containing a nonempty set has an element contained in at least half of its members. Let $c_{\mathrm{UC}}$ denote the largest universal lower bound on this proportion. The best previously proved lower bound is due to Liu and is above $0.3823455$; his own numerical optimization suggested a stronger threshold $c_{\mathrm L}>0.382709087918735$. The main argument of this paper is to use different protocols (suitable families of probability measures) within Liu's framework. In particular, we combine the independent protocol with Liu's Example 5 to prove the conjectured bound $c_{\mathrm{UC}}\ge c_{\mathrm L}$. We then vary the parameter in Example 5 and optimize the choice of the two coefficients in the combination to obtain $c_{\mathrm{UC}}\ge 0.3828852549667978$. We also show that the inequality used in this proof cannot give a threshold larger than $0.3828852599667\ldots$.

We prove that the maximal cp-rank $p_n$ satisfies $p_n\ge n(n-3)/2$ for odd $n\ge5$ and $p_n\ge n(n-3)/2-1$ for even $n\ge6$. Together with the known upper bound $p_n\le\binom{n+1}{2}-4$, this gives $p_n=n^2/2+O(n)$.

For edge-disjoint graphs meeting along a finite boundary, we construct relative shellings of NBC faces with prescribed boundary connectivity. The construction adds paths in spanning forests to the restriction faces of auxiliary rooted-forest complexes. When every edge of the first graph precedes every edge of the second, these relative shellings combine into a shelling with consecutive connectivity blocks. The resulting nonnegative decomposition of the $h$-polynomial realizes the chromatic specialization of the Negami splitting identity. We also give path formulas for gluing graphic Orlik–Terao Gröbner bases across boundaries of size two or three, and a virtual-clique elimination formula for arbitrary boundaries. An exponential output lower bound at treewidth two distinguishes explicit basis enumeration from coefficient computation at fixed treewidth. Finally, state polytopes and their Cayley lifts describe an obstruction to linear block orders in the original coordinates and a lifted realization of these orders.

Choose a uniformly random ordering of the vertices of a finite tree and run the usual greedy maximal-independent-set algorithm. We compare the resulting law on maximal independent sets with the uniform law. We prove that exact uniformity occurs only for the one-vertex tree and the single edge. The proof is structural: a diameter endpoint exposes a pendant star, and the remaining one-pendant-leaf case is resolved by a strict injection between exact permutation fibres obtained by swapping the pendant leaf with its support vertex. Exact uniformity is therefore rigid, but it can be approached closely. For an explicit mixed-spider family $T_{k,l}$ we count the maximal independent sets and compute the exact probability of every output. With $l=2^k-k$ the total-variation bias is positive and satisfies [ b(T_{k,2^k-k})=O!\left(\frac{\sqrt{k}}{4^k}\right) =O!\left(\frac{\sqrt{\log n_k}}{n_k^2}\right), \qquad n_k=2^k+k+1. ] The theorem package has also been formalised in Lean and registered with Palomar. These records document machine-checked formal verification and the checked axiom boundary; they are not peer review or a certificate of novelty.

An edge open packing is a set of edges whose endpoints induce a disjoint union of stars. We study the problem of finding a maximum edge open packing in AT-free, chordal, and convex bipartite graphs. For AT-free graphs, we use the oriented star-conflict graph $A_G$, introduced by Das and Santra, for which $\rho_e^o(G)=\alpha(A_G)$, and prove that $A_G$ is AT-free whenever $G$ is AT-free. Consequently, a known maximum independent set algorithm for AT-free graphs yields an $O(n^2+m^4)$-time algorithm for Maximum Edge Open Packing, where $n$ and $m$ denote the numbers of vertices and edges of $G$, respectively. For chordal graphs, we develop an $O(n^4)$-time dynamic programming algorithm over a nice tree decomposition derived from a clique tree, exploiting the structural fact that the endpoint set of an edge open packing intersects every clique in at most two vertices. Finally, for convex bipartite graphs, we obtain an $O(n^4)$-time dynamic programming algorithm based on two boundary indices that separate consecutive star components. This result extends the previously known algorithm for biconvex bipartite graphs and, in particular, provides an improved explicit running-time bound for that class.

Let $G$ be an $n$-vertex connected nonregular graph with maximum degree $\Delta$ that attains the maximum spectral radius. For $3\le\Delta\le n-2$, Liu and Li (2008) conjectured that $G$ has the degree sequence $(\Delta,\ldots,\Delta,\delta)$ with $\delta<\Delta$ as large as possible. In this paper, we study the dense case where $\Delta=n-k$ with $k\ge4$ fixed and $n$ sufficiently large. We prove that \[ d(G)= \begin{cases} (\Delta,\ldots,\Delta, 1), & \text{if $n$ is even and $k$ is odd};\\ (\Delta,\ldots,\Delta,\Delta-1), & \text{if $n$ is odd and $k$ is even};\\ (\Delta,\ldots,\Delta,2), & \text{if $n$ and $k$ have the same parity}. \end{cases} \]

A prime labeling of a graph labels its vertices bijectively with the integers from one to the number of vertices, so that adjacent vertices receive coprime labels. I prove the Entringer-Tout conjecture that every tree admits a prime labeling, extending the result of Haxell, Pikhurko, and Taraz from sufficiently large trees to all orders. My proof combines structural decompositions of trees, arithmetic organization of the labels, and matching arguments. I reduce the labeling problem to a structured bipartite matching problem by separating a small part of the tree, distributing labels according to their divisibility properties, and applying Hall's theorem to complete the labeling. I handle the finite and intermediate ranges using exact computer-assisted certificates whose validity is justified mathematically, while the remaining range is covered by analytic estimates. Together, these ingredients yield a proof for trees of every order.

For a positive integer $n$, let $S(n)=\{\sigma(d): d\mid n\}$ be ordered by divisibility, and let $a(n)$ be its width. We study how much of the divisor lattice of $n$ survives under the map $d\mapsto \sigma(d)$. For every fixed prime-exponent pattern, the largest possible value of $a(n)$ is the width of a corresponding product of divisibility posets, and this maximum is attained infinitely often. In the squarefree case with $\omega(n)=r$, every width from $1$ to $\binom{r}{\lfloor r/2\rfloor}$ occurs, again infinitely often. For repeated exponent patterns, Engel's limit theory reduces the asymptotic width problem to optimal representation variance. We show that for the divisibility poset $D_m=(\{1,\ldots,m\},\mid)$, the prime directions $2$ and $3$ already determine the exact variance. The structural reason is the critical dilation $4$, lying strictly between $3$ and the next prime $5$. This yields an exact finite formula, the limit $V(D_m)\to 1.0367934201598\ldots$, and asymptotics for repeated exponent patterns. The complementary minimum-width problem remains largely open.

A binary word is a shuffle square if its positions can be partitioned into two identical subsequences. A finite word over $\{1,2\}$ is smooth if repeated finite run-length differentiation remains binary until the empty word is reached. This paper proves that every smooth binary word in which both letters have even multiplicity is a shuffle square once its length is at least $18$. There are exactly $34$ exceptions, all of length at most $16$. The infinite part of the proof is a parity-return atom induction: a smooth even-Parikh word factors uniquely into smooth parity atoms, and every such atom has length at most $8$. A finite, independently checkable certificate handles lengths $18$ through $40$. As a consequence, a nonempty prefix of the classical Kolakoski sequence is a shuffle square if and only if both symbols occur an even number of times and its length is not $4$ or $8$. There are infinitely many such prefixes. In fact, the entire infinite Kolakoski sequence can be partitioned into two identical infinite subsequences with monochromatic blocks of uniformly bounded length.

In 1993, Ruzsa showed that for every $k \geq 2$, there exists a constant $C$ such that every subset $A \subseteq [N]$ of size at least $C N^{1/k}$ contains $2k$ distinct elements $a_1, \ldots, a_k, b_1, \ldots, b_k \in A$ such that $a_1 + \cdots + a_k = b_1 + \cdots + b_k$. We strengthen this result by proving that the elements $a_1, \ldots, a_k, b_1, \ldots, b_k$ can be chosen to have the additional property that $\{a_1, \ldots, a_k, b_1, \ldots, b_k\}$ has $2^{2k}-1$ distinct subset sums, with the only coincidence being that $\{a_1, \ldots, a_k\}$ and $\{b_1, \ldots, b_k\}$ have the same sum. Our proof also applies to any finite Abelian group of odd order $N$, and it provides a corresponding supersaturation result: that whenever $|A| \geq CN^{1/k}$, there are at least $\Omega(|A|^{2k}/N)$ choices for $a_1, \ldots, a_k, b_1, \ldots, b_k \in A$ satisfying these properties. We prove a slightly weaker statement for Abelian groups of even order. We also apply our methods to the vector space setting and prove the following $\mathbb F_q$-analogue of the Bondy-Simonovits Theorem on the extremal number of even cycles in graphs: Any rank-$n$, simple, $\mathbb F_q$-representable matroid with no circuit of size exactly $2k$ has size at most $C q^{n/k}$ for some constant $C$ depending only on $q$ and $k$. When $q=2$, this is best possible up to the constant $C$ for all $k \geq 2$. Our methods also apply to the original graph setting and give a new proof of the Bondy-Simonovits Theorem and its supersaturation version.

What is the maximum number of rainbow triangles in an edge-colored graph with $m$ edges and $r$ colors? Using entropic techniques, we prove an upper bound of $C_rm^{3/2}$ rainbow triangles with $C_r=\sqrt{\frac{2(r-2)}{9r}}$; this constant is best possible whenever there exists an affine plane of order $r-1$. We also show that constructions attaining at least $(C_r-\varepsilon_r)m^{3/2}$ rainbow triangles must exhibit an affine plane structure, which further improves the upper bound if no such affine plane exists. We also consider the problem of counting properly edge-colored cliques of larger sizes. Surprisingly, if the number $r$ of colors is odd, this count is instead maximized by blowups of a properly edge-colored $K_{r+1}$.

We present a version of the vector balancing problem in which each vector may be given a sign and a permutation of its coordinates. We prove that this vector balancing problem and its corresponding prefix problem admit an explicit bound, and we further show that it is asymptotically optimal in the dimension. Our method of proof is purely geometric.

A family of sets is intersecting if every two members intersect, and trivial if all its members contain a common element. We determine the typical structure of $k$-uniform intersecting families on $2k+1$ and $2k+2$ elements as $k\to\infty$. For $n=2k+2$, we prove that almost all intersecting families are trivial and that their number is \[ (2k+2+o(1))\,2^{\binom{2k+1}{k-1}}. \] Together with Yang's recent result for $n\ge 2k+3$, this settles a conjecture of Balogh, Garcia, Li, and Wagner. For $n=2k+1$, almost all intersecting families are nontrivial. We prove that, as conjectured by the same authors, a typical intersecting family is close to a full star: its members outside the star form components of size at most two in the graph joining sets that intersect in $k-1$ elements. We also obtain an asymptotic formula for the number of intersecting families in this case, with an explicit second-order term in the exponent. Our proof combines Sapozhenko's graph container method and stability in Kneser graphs to control families far from every star, and a polymer model and cluster expansion to enumerate families close to a fixed star.

The linear arboricity $\la(G)$ of a graph $G$ is the least cardinality of linear forests, that is, forests of maximum degree at most $2$, into which its edge set $E(G)$ can be decomposed. The Linear Arboricity Conjecture asserts that $\la(G)\leq\lceil(\Delta(G)+1)/2\rceil$ for every finite graph $G$, where $\Delta(G)$ denotes the maximum degree of $G$. We extend this conjecture to infinite graphs of finite maximum degree and prove that its finite and infinite versions are equivalent. We introduce topological linear arboricity $\latop (G)$ by requiring the linear forests to contain no topological circle, and show that it differs from linear arboricity by at most one. Finally, we prove that every $2k$-regular graph of girth at least $2k$ has topological linear arboricity at most $k+1$.

We prove that there is an absolute constant $c>0$ such that every connected vertex-transitive graph $G$ on $n \ge 3$ vertices contains a cycle of length at least $cn$. Moreover, every such graph with sufficiently large degree $d$ contains a cycle of length at least $(1-d^{-1/100})n$. This gives the first linear bound towards Lovász's Hamiltonicity conjecture. The proof combines a structural result of DeVos and Mohar on vertex-transitive graphs with a general framework for finding long cycles, which may be of independent interest.