Papers

Let $T$ be a tree. For a vertex $v\in V(T)$, the eccentric subtree number $\epsilon_T(v)$ is defined as $\epsilon_T(v)=\min\{f_T(v,u): u\in V(T)\}$ where $f_T(v,u)$ denotes the number of subtrees of $T$ containing both $v$ and $u$. A core vertex of $T$ is a vertex with the maximum eccentric subtree number, and the set of all the core vertices of $T$ is called the core center of $T$. The core center of $T$ consists of either a single vertex or two adjacent vertices. There are other central concepts in a tree, such as the center, centroid, subtree core, and characteristic center, and these may all be different. By $d_T(C, \mathfrak{C})$, $d_T(C_d, \mathfrak{C})$ and $d_T(S_c, \mathfrak{C})$ we mean the distance between center and core center, distance between centroid and core center and distance between subtree core and core center in $T$, respectively. We show that for any tree $T$ on $n\geq 6$ vertices, (i)]$d_T(C,\mathfrak{C})\leq \lfloor \frac{n-g_0-4}{2} \rfloor$; (ii)]$d_T(C_d,\mathfrak{C})\leq \lfloor \frac{n-5}{2} \rfloor$; (iii)] $d_T(S_c,\mathfrak{C})\leq\left\{ \begin{array}{ll} 1, &\text{if $n=7$,} n-g_0-3, &\text{if $n\neq 7$;}\\ \end{array} \right.$ where $g_0\geq 2$ be the smallest positive integer such that $2^{g_0-1}+g_0\geq n-3$. Moreover, we show that these bounds are best possible by obtaining a tree which attains these bounds. We also obtain a tree which maximizes the distance between characteristic center and core center over all trees on $n\geq 6$ vertices. The asymptotic behaviour of all these distances are also studied.

This paper studies the giant component of the sparse random bipartite graph $G(n, n, p)$, where $p = c/n$ for a fixed constant $c > 1$. We prove that its size is asymptotically normal.

We prove that the filtered Varchenko–Gelfand algebra is invariant under a natural Cremona operation on a class of real hyperplane arrangements. Namely, suppose that an arrangement contains all coordinate hyperplanes and that every remaining defining form is supported on two coordinates. Swapping the two coefficients in each such form produces its Cremona transform. Coordinatewise inversion gives a chamber bijection and an isomorphism of the corresponding filtered Varchenko–Gelfand algebras over every commutative coefficient ring. As an application, we exhibit two arrangements of eight central planes in $\mathbb{R}^3$ with isomorphic filtered Varchenko–Gelfand algebras but non-isomorphic tope graphs. This disproves a conjecture of Yagi–Yoshinaga on reconstructing tope graphs from filtered Varchenko–Gelfand algebras.

Deodhar diagrams give a combinatorial way to compute point counts of open positroid varieties over finite fields. We introduce affine Deodhar diagrams, which extend this construction to affine patches of open positroid varieties. These diagrams are indexed by a bounded affine permutation together with a lattice path, and this extra flexibility makes them especially useful for recursive bijections. Motivated by connections with Dyck paths, open positroid varieties, and their cluster structure, we construct bijections between several classes of affine Deograms. These bijections give combinatorial proofs of point-count identities that were previously known from geometric isomorphisms.

A balanced $k$-partite $k$-graph is a $k$-uniform hypergraph such that every edge intersects each partition class in exactly one vertex, where each partition class has size $n$. Lo and Markström (2014) determined the minimum vertex-degree threshold for perfect matchings in balanced \(3\)-partite \(3\)-graphs. In this paper, we determine the minimum vertex-degree threshold for balanced \(4\)-partite \(4\)-graphs. The proof relies on a reduction framework for \(k\)-partite \(k\)-graphs, through which the existence of a perfect fractional matching is converted into a finite-dimensional optimization problem.

For a fixed $r$, let $f_r(n)$ denote the minimum number of complete $r$-partite $r$-uniform hypergraphs whose edge sets partition the complete $r$-uniform hypergraph on $n$ vertices. The Graham-Pollak theorem asserts that $f_2(n)=n-1$. Let $c_r$ be the smallest constant such that $f_r(n)\le c_r(1+o(1))\binom{n}{\lfloor r/2\rfloor}$. We give two constructions that improve upper bounds for $f_r(n)$. The first partitions $E(K_4)\times E(K_{17})$ into $44$ products of edge sets of complete bipartite graphs and yields $c_4\le11/12$, improving the previous bound of $c_4\le14/15$. The second gives an asymptotically improved exact cover for odd $r$. Combining these constructions, we prove that $c_r<1$ for every odd integer $r\ge65$, which improves upon the previous computer-assisted result of $113$. We further obtain the improved asymptotic bound $c_r\le \frac{r}{15}\left(\frac{11}{12}\right)^{r/4}+o(1)$ for general $r$. These results narrow the gap towards resolving the major open problem of determining whether $c_5<1$.

Fuglede's conjecture asserts that a measurable set of positive and finite measure is spectral if and only if it tiles Euclidean space by translations. Counterexamples are known in every dimension $d\ge3$, whereas the one- and two-dimensional cases have remained unresolved. We construct two explicit $60$-point subsets of the rank-two finite Abelian group $\Z_{60}\times\Z_{12}$: one is a translational tile with no spectrum, and the other is spectral but does not tile. A finite-to-infinite transference principle lifts them to bounded subsets of $\R^2$ that are finite unions of unit squares. Consequently, both implications in Fuglede's conjecture fail in dimension two.

Standard phylogenetic reconstruction techniques often yield unrooted phylogenetic networks; these are subsequently rooted to infer evolutionary history. A common problem in this process is to determine the structural classes to which the resulting network will belong. In this paper, we investigate unrooted networks in which the choice of any root results in a valid rooted phylogenetic network, a property we define as robustly orientable. We then establish a strict structural condition for this class, specifically, that an unrooted network is robustly orientable if and only if it contains no sink components. We also show that if an unrooted network is level-$2$ or less, or if it is tree-based, then it is robustly orientable. Furthermore, we define an unrooted network to be {\em robustly class $\mathcal C$} if the choice of any root results in a network belonging to class $\mathcal C$. We demonstrate that an unrooted network is robustly tree-child or robustly stack-free if and only if it is level-$1$ or less. Finally, we show that a phylogenetic network is robustly normal if and only if it is a phylogenetic tree.

For a 1-factorization $F=\{M_1,\ldots,M_d\}$ of the hypercube $Q_d$, let $G[F]$ have vertex set $F$, with $M_iM_j$ an edge exactly when $M_i\cup M_j$ is a Hamilton cycle. Behague proved that $Q_{k+\ell}$ has a 1-factorization $F$ with $G[F]\cong K_{k,\ell}$ for all positive $k,\ell$ except possibly $k=\ell=3$. We give an explicit 1-factorization of $Q_6$ for which $G[F]\cong K_{3,3}$, resolving the exceptional case. The construction is supplied as a finite certificate. Its correctness can be checked directly from the tables in the paper or by either of two independent, short, standard-library verifiers supplied with the certificate.

Gupta and Iosevich introduced the edge complexity of a graph as the minimum Fourier ratio of its adjacency matrix over all vertex labelings and bounded it below by graph energy divided by the square root of twice the number of edges. We characterize equality for a fixed labeling: the Fourier transform of the adjacency matrix must have at most one nonzero entry in each row and column. This implies regularity, circulancy of every positive even power of an extremizing adjacency matrix, and a parity restriction on connected components, and it gives equality results for certain Laplacian spectral projectors. We construct equality cases from affine involutions on cyclic groups. Singer difference sets yield, for every prime power $q$, an equality-attaining $(q+1)$-regular graph that is not an abelian Cayley graph. We also establish Fourier-ratio estimates for weak, Cartesian, and strong graph products, including preservation of equality under weak products of coprime orders. We use Fourier-ratio recovery as a coding theorem to obtain entropy upper bounds for low-complexity adjacency matrices and complement them with a lower bound obtained by perturbing complete graphs. Finally, a concentration argument shows that if $Np_N/\log N\to\infty$ and $\limsup_{N\to\infty}p_N<1$, then $\operatorname{FR}_{\min}(G(N,p_N))$ is of order $N$ with probability tending to one.

Let $R$ be a ring with identity. The generalized von Neumann inverse graph of $R$, denoted by $\Gamma_{Reg}(R)$, is defined as the graph whose vertex set is $Reg(R)$, where two distinct vertices $a,b \in R$ are adjacent if and only if $aba=a$ or $bab=b$. In this work, we consider the reduced graph $\Gamma'_{Reg}(R)$ obtained by restricting the vertex set to $Reg(R)\setminus{0_R}$, so that $\Gamma_{Reg}(R) \cong K_1 + \Gamma'_{Reg}(R)$, allowing the analysis to focus on its nontrivial structure. We investigate the structure of $\Gamma'_{Reg}(R)$ for finite commutative von Neumann regular rings and establish several results describing its graph-theoretic properties in relation to the algebraic structure of $R$. In particular, we derive conditions that characterize connectivity, acyclicity, and planarity, and examine structural features such as vertex degrees, girth, and the existence of pendant vertices, along with their algebraic implications. We also identify circumstances under which $\Gamma'_{Reg}(R)$ exhibits specific graph classes, including paths, cycles, and wheels, as well as the presence of certain induced subgraphs. Furthermore, an explicit algorithm is provided to construct $\Gamma'_{Reg}(R)$, and connections with the inclusion ideal graph of $R$ are discussed, offering additional insight into the interplay between ring-theoretic properties and graph structures.

The recent breakthrough work of Chatterjee, Ghosh, Gurjar, Raj and Thierauf [CGGRT26] gives the first deterministic NC algorithm for the bipartite matching problem. They show how to detect as well as find perfect matchings in bipartite graphs in NC. In this note we present an arguably simpler-to-state variation of the NC detection criterion of [CGGRT26], with improved parameters.

We refine Radhakrishnan's entropy proof of the Brégman-Minc bound by introducing a terminal-set framework in which selected vertices are revealed last. This gives new degree-sensitive upper bounds for the number of perfect matchings in bipartite graphs and an explicit formula for single-vertex terminal sets. The bounds recover the standard Brégman equality family and improve the estimate for certain nonuniform degree sequences. We also obtain a $C_4$-free refinement complementary to the edge-count bound of Araujo, Balogh and Wang.

We prove that every connected simple graph of bounded degree satisfying the classical dimension-free Bakry–Émery condition $\mathrm{CD}(0,\infty)$ for the unnormalised Laplacian is volume doubling and supports, at all integer graph scales, a scale-invariant $L^2$-Poincaré inequality with dilation two, with constants depending only on the maximum degree. This settles the polynomial-growth conjecture of Cushing, Liu, and Peyerimhoff in a stronger form. The main novelty is a dimension-free adaptation of the graph-theoretic modified nonlinear heat-flow method introduced by Münch and extended to infinite weighted graphs by Pajot and Russ: point-mass consequences of $\Gamma_2\geq0$ and positive-resolvent smoothing replace any global $\mathrm{CD}(0,n)$ reduction, while diffusive exit-time control and finite-volume localisation yield the Poincaré inequality.

Bilu and Linial conjectured that if $d\ge 2$, then every $d$-regular graph $G$ has an edge signing $\sigma:E(G)\to\{-1,1\}$ such that its signed adjacency matrix $A_\sigma$ satisfies the Ramanujan bound \begin{equation*} \rho(A_\sigma)\le 2\sqrt{d-1} \end{equation*} and they proved that \begin{equation*} \rho(A_\sigma)=O\!\left(\sqrt{d\log^3 d}\right). \end{equation*} Using a method of interlacing polynomials, Marcus, Spielman, and Srivastava confirmed one side of this conjecture that there is a signing $\sigma$ for which \[ \lambda_{\max}(A_\sigma)\le 2\sqrt{d-1}. \] By constructing an auxiliary bipartite graph from a balanced orientation of $G$ and applying a bipartite signing argument, we prove that every finite simple graph $G$ of maximum degree $d\ge 3$ admits an edge signing $\sigma$ such that \[ \rho(A_\sigma) \le 4\sqrt{\left\lceil \frac d2\right\rceil-1} \le 2\sqrt{2(d-1)}. \] Thus we obtain a bound within a factor at most $\sqrt2$ of the conjectured Ramanujan bound.

This work is part of a research program to compute the Hochschild homology groups $\text{HH}_*(\mathbb{C}[x_1, ..., x_d]/(x_1, ..., x_d)^3; \mathbb{C})$ via Coxeter cohomology, which utilizes the isomorphism \[ \text{HH}_i(\mathbb{C}[x_1, ..., x_d]/(x_1, ..., x_d)^3; \mathbb{C}) \cong \sum_{0 \leq j \leq i} H_C^j\bigg( S_{i+j}, V^{\otimes(i+j)} \bigg)\] provided by Larsen and Lindenstrauss. Here, $H^*_C$ denotes Coxeter cohomology, $S_{i+j}$ is the symmetric group on $i+j$ letters, and $V$ is the standard representation of $\text{GL}_d(\mathbb{C})$ on $\mathbb{C}^d$. While previous work has focused exclusively on the case $d=2$, we extend some results to all values of $d$. Notably, we show that when the tensor representation is replaced by an irreducible representation, the Euler characteristic of the corresponding Coxeter cohomology is given by a polynomial with bounded degree. Although the problem is motivated by algebra and topology, the solution relies primarily on combinatorial arguments.

We introduce a Tutte polynomial for hypergraphs, $T_{\mathrm{HG}}$, together with $T_k$, a related Tutte polynomial for $k$-polymatroids. Both invariants admit deletion–contraction recursions that remain within their respective classes, and they are linked by the fact that $T_{\mathrm{HG}}$ specializes to $T_k$ on the associated polymatroid of any $(k+1)$-uniform hypergraph. We show that $T_{\mathrm{HG}}$ satisfies several desirable Tutte type properties, including multiplicativity and duality, while $T_k$ further admits a universality theorem, as well as a convolution product formula. In the uniform hypergraph setting, these latter results specialize back to $T_{\mathrm{HG}}$. We also relate $T_{\mathrm{HG}}$ to hypergraph extensions of the Potts and random cluster models. In particular, we study degree dependent random cluster and Potts partition functions, as well as Grimmett's many body Potts model, and compare their relationship with $T_{\mathrm{HG}}$ in both the general and uniform settings. Finally, we compare $T_{\mathrm{HG}}$ with the polymatroid Tutte polynomial $\mathcal{T}_P$ of Bernardi, K'alm'an, and Postnikov, showing that the two are incomparable in distinguishing power. As a consequence, we answer negatively a question raised by these authors by proving that the characteristic polynomial is not, in general, a specialization of $\mathcal{T}_P$.

For $\mathbf{b}=(b_1,\dots,b_n)\in\mathbb{Z}_{>0}^n$, a $\mathbf{b}$-parking function is a sequence $(\beta_1,\dots,\beta_n)$ of positive integers whose nondecreasing rearrangement $\beta_1'\le\beta_2'\le\cdots\le\beta_n'$ satisfies $\beta_i'\le b_1+\cdots+b_i$. The $\mathbf{b}$-parking-function polytope $\mathfrak{X}_n(\mathbf{b})$ is the convex hull of all $\mathbf{b}$-parking functions of length $n$ in $\mathbb{R}^n$. We prove that every lattice slice of $\mathfrak{X}_n(\mathbf{b})$, obtained by fixing one coordinate at an integer value, is itself a $\mathbf{b}'$-parking-function polytope of one dimension less, with an explicit parameter vector $\mathbf{b}'$; this yields a recursion for the number of lattice points of $\mathfrak{X}_n(\mathbf{b})$. We further show that every dilate of a $\mathbf{b}$-parking-function polytope is a translate of another such polytope, that the number of lattice points is a polynomial function of $\mathbf{b}$, and we deduce an explicit formula for the Ehrhart polynomial of $\mathfrak{X}_n(\mathbf{b})$ for arbitrary $\mathbf{b}$ as a finite sum indexed by draconian sequences, resolving a problem of Hanada, Lentfer, and Vindas-Meléndez; an equivalent formula was recently obtained, independently, by Liu and Thawinrak in a closely related setting. In the special case $\mathbf{b}=(a,b,\dots,b)$, we obtain an explicit closed form and a generating function for the Ehrhart polynomial. As an application, we classify magic positivity in the two-parameter family $\mathfrak{X}_n(a,b)=\mathfrak{X}_n(a,b,\dots,b)$: the polytope $\mathfrak{X}_n(a,b)$ is magic positive if and only if $(n,a,b)\ne(2,1,1)$. Thus, we answer a problem posed by Ferroni and Higashitani for $\mathfrak{X}_n(a,b)$. Our result extends recent work of Liu and Zhang on partial permutahedra and leads us to conjecture that magic positivity holds for every $\mathfrak{X}_n(\mathbf{b})$ with $n\ge3$.

Fix integers $0\leq k<n$. Let $F_{n,k}(N)$ be the least size of a finite set $B\subset\Z^n$ that contains a filled axis-parallel cube $k$-skeleton centered at each point of some $N$-point set. We prove that $F_{n,k}(N)$ has order $N^{1-(n-k)/(2n^2)}$, with constants depending only on $n$ and $k$. Thornton proved the upper bound and lower bounds with every smaller exponent; the endpoint lower bound was open for $k\geq1$. For square boundaries in $\Z^2$, the exponent is $7/8$. A midpoint count and Shearer's inequality handle large radii; induction in lattice cells handles small radii.

We mainly consider three problems on cycle lengths and cycles with chords in graphs: (a) Gao, Huo, and Ma \cite[Question~1.5]{GaoHuoMa2021} asked whether, for every fixed $k\ge3$, there is a function $f_k(n)\to\infty$ such that every $n$-vertex $(k+1)$-critical graph contains $f_k(n)$ consecutive cycle lengths. (b) Let $g_k(n)$ be the maximum integer $t$ such that every $n$-vertex $k$-critical graph with $k\ge4$ contains an odd cycle with at least $t$ chords. Voss conjectured (see \cite[pp.~168]{VossBook}) that $g_k(n)\to\infty$ as $n\to\infty$ for each $k\ge4$, which extends a 1976 conjecture of Erdős (see also Erdős Problem~1091 \cite{Bloom1091}). (c) Kára and Král \cite{KaraKral2003} conjectured that every graph on $31$ vertices with minimum degree at least $8$ contains a cycle with at least $31$ chords. We answer question (a) in the negative for $k=3$, and disprove conjecture (b) for all $k\ge5$. We point out the work of Alexeev-Putterman-Sawhney-Sellke-Valiant (2026) on Erdős Problem 1901 disproves the case $k=4$ for conjecture (b). We prove conjecture (c). We also discuss two other related problems in the part of concluding remark.

Fix a constant $c$ with $0<c<1$. We say a graph $G$ on $n$ vertices is $c$-hollow if every maximal independent set of $G$ has size at least $cn$. Denote by $\tau(G)$ the size of a smallest set of vertices $T\subseteq V(G)$ such that every maximal independent set in $G$ intersects $T$, i.e., $T$ is a transversal for the family of maximal independent sets. In 1991, Bollobás, Erdős, and Tuza conjectured that if $G$ is $c$-hollow, then $\tau(G)=o(n)$. Using a random construction, we show there exist $c$-hollow graphs with $\tau(G)=\Omega\left(\frac{n^{1/3}}{\log n }\right)$, establishing the first nontrivial lower bound constraining the conjecture and complementing a closely related lower bound due to Alon for maximum independent sets. We also show the conjecture holds in a strong form for the class of cographs and split graphs.

We show that there is a constant $c>0$ such that, for all sufficiently large $N$, there is a subset $A \subseteq \{1,\dots,N\}$ of size $>cN$ such that for any two distinct elements $a,b$ in $A$, the average of $\frac{1}{a}$ and $\frac{1}{b}$ is not a unit fraction, negatively answering a question of Erdős and Graham. This also gives the best known lower bounds on the maximum size of a set of unit fractions without non-trivial three-term arithmetic progressions.

The following notes are my attempt to clarify, to myself, the OpenAI proof of the Cycle Double Cover Conjecture. This write-up offers no new insights, but some people may find it easier to read.

We study the minimum arclength of spherical $t$-design curves, i.e., closed rectifiable curves on $S^d$ whose normalized arclength measure exactly integrates every polynomial of degree at most $t$. We prove an explicit spectral lower bound that is sharp for $t=1$ in all spheres and for $t=2$ in every odd-dimensional sphere, yielding the first exact optimality results for spherical $t$-design curves with $t>1$. For even-dimensional spheres, we construct $2$-design curves whose lengths asymptotically match the lower bound as $d\to\infty$, and in $S^2$, we use numerical optimization and the calculus of variations to derive a candidate for the shortest $2$-design curve.

Brändén and Huh showed that Lorentzian polynomials unify Hodge-Riemann relations in combinatorics: their supports are M-convex, and every M-convex set supports a Lorentzian polynomial. Baker, Huh, Kummer, and Lorscheid later proved that, for every $q>0$, the projectivized space $\mathbf{P}\operatorname{L}_J$ of Lorentzian polynomials with support $J$ is homeomorphic to the thin Schubert cell $\operatorname{Gr}^{\mathrm{w}}_J(\mathbb{T}_q)$ of weak representations of $J$ over the generalized triangular hyperfield $\mathbb{T}_q$. We study the quantitative relation between Lorentzian polynomials and representations over triangular hyperfields. For every matroid $M$, we prove that some $q>0$ depending on $M$ satisfies $\operatorname{Gr}^{\mathrm{w}}_M(\mathbb{T}_q)\subseteq\mathbf{P}\operatorname{L}_M\subseteq\operatorname{Gr}^{\mathrm{w}}_M(\mathbb{T}_2)$. Thus $\mathbf{P}\operatorname{L}_M$ lies between two thin Schubert cells, each homeomorphic to it. More generally, for every M-convex set $J$, some $q>0$ depending on $J$ satisfies $\operatorname{N}\operatorname{Gr}^{\mathrm{w}}_J(\mathbb{T}_q)\subseteq\mathbf{P}\operatorname{L}_J\subseteq\operatorname{N}\operatorname{Gr}^{\mathrm{w}}_J(\mathbb{T}_2)$, where $\operatorname{N}$ denotes normalization. We also study $q(M):=\sup\{q>0:\operatorname{Gr}^{\mathrm{w}}_M(\mathbb{T}_q)\subseteq\mathbf{P}\operatorname{L}_M\}$. For $q(n):=q(U_{2,n})$, we prove $q(4)=2$ and $q(5)=\log_2 3$, with matching upper and lower bounds of order $1/n$; hence $q(n)=\Theta(1/n)$, so in particular no universal positive lower bound for $q(n)$ exists.

We prove that the longer and shorter Sackin indices of a uniformly random simplex tree-child network with $n$ taxa admit joint distributional limits after rescaling by $n^{-7/4}$. The limiting distributions are described by functionals of a Brownian excursion. We also identify the limiting law of the height after rescaling by $n^{-3/4}$, thereby answering a question of Zhang~(2022). Moreover, we establish sharp tail bounds for the height, which imply convergence of all moments in the above distributional limits. We further obtain a scaling limit for the entire height profile of the leaves. Finally, we determine the local limits of large simplex networks around the fixed root, a uniformly random vertex, and a uniformly random leaf.

The snake-in-the-box problem asks for a longest induced path in the hypercube graph $Q_n$. We find a length-191 snake in dimension $n=9$, the lowest dimension where the maximum is unknown, improving the previous record of 190 that had stood for 14 years. We also establish new lower bounds in dimensions 10-13. To find these records, we introduce snakepits, collections of disjoint snakes, to expand the search space and open new routes between snakes. This motivates our new Snakepit-in-the-Box benchmark, which seeks maximal edge counts when allowing multiple components. Finally, we introduce Beam Anchor, a search-supervised learned constructor algorithm that finds 100 inequivalent length-190 snakes in dimension 9.

We prove, for all fixed $0 < \delta < 1$, and all sufficiently large $n$, that there exists $S \subset [n]$ with $|S| \ge \delta n$ such that $A + B \not \subset S$ for all ${A, B \subset \mathbb{N}}$ satisfying $$\min\big\{|A|, |B|\big\} \ge \big(3 + o(1)\big) \frac{\log n }{ \log (1 / \delta)}.$$ A very recent result of Hernández and Hetzel shows that our bound is sharp up to a factor of 3, and together our results settle a conjecture of Kra, Moreira, Richter, and Robertson. In fact, we prove that a $\delta$-dense random subset of $[n]$ is a valid choice for $S$ with high probability, and that one can take $n^{-\alpha} \le \delta \le 1 - c$ where $c > 0$ is fixed and $\alpha > 0$ depends only on the $o(1)$ error, answering another question of the same authors in a strong form.

A set of points $C \subset \mathbb{R}^n$ is called canonically Ramsey if there is some set of points $S\subset \mathbb{R}^{n'}$ such that any colouring of $S$, using any number of colours, must contain either a monochromatic copy of $C$ or a rainbow copy of $C$. Mao, Ozeki, and Wang introduced this notion, showing that 30-60-90 triangles are canonically Ramsey. Since then, various other canonically Ramsey configurations have been identified. The author showed that cuboids are canonically Ramsey, while Ge, Shu, Xu, and Yu recently showed that simplices are canonically Ramsey. We extend both of these results, proving that all products of simplices are canonically Ramsey.

In this paper, we establish an analogue of the Kővari-Sós-Turán Theorem for $\operatorname{GF}(q)$-representable matroids. For $2\leq s\leq t$, we show that if $M$ is a rank-$n$ simple $\operatorname{GF}(q)$-representable matroid having no $M(K_{s,t})$-restriction, then \[ |E(M)|=O_{q,s,t}\bigl(q^{(1-1/s)n}\bigr). \] In particular, we prove that the maximum number of elements in a simple rank-$n$ binary matroid with no $M(K_{2,t})$-restriction is $\Theta_{t}(2^{n/2})$ where the lower bound is obtained using binary Sidon sets.