Papers

We study probability distributions on independent sets of uniform hypergraphs with local constraints. For a positive vertex-weight function $w$ on an $(r+1)$-uniform hypergraph, define the weighted degree of vertex $v$ by \[ d_w(v):= \sum_{e\ni v} \left( \prod_{u\in e\setminus\{v\}}\dfrac{w(u)}{w(v)} \right)^{1/r}. \] For each fixed $r\geq1$, our first result gives, in every triangle-free $(r+1)$-uniform hypergraph (without a linearity assumption), a random independent set $I$ satisfying \[ \mathbb P(v\in I)\geq \left( \frac{r}{r+1}(r(r+1))^{-1/r}+o_r(1) \right) \left(\frac{\log d_w(v)}{d_w(v)}\right)^{1/r}, \qquad d_w(v)\to+\infty. \] The same result also implies that every $d$-degenerate triangle-free $(r+1)$-uniform hypergraph satisfies \[ \chi_f(G)\leq \left( \dfrac{r+1}{r}(r(r+1))^{1/r}+o_r(1) \right) \left(\dfrac{d}{\log d}\right)^{1/r}, \qquad d\to+\infty. \] Our second result concerns linear Berge-$C_4$-free hypergraphs which allow Berge triangles. In this setting, for every sufficiently large threshold $D$, there exists a random independent set $I$ such that, uniformly over all vertices with $d_w(v)\geq D$, \[ \mathbb P(v\in I)\geq (1-o_r(1)) \left(\frac{\log d_w(v)}{r\,d_w(v)}\right)^{1/r}, \qquad D\to+\infty. \] It also yields \[ \chi_f(G)\leq (1+o_r(1)) \left(\frac{rd}{\log d}\right)^{1/r} \] for $d$-degenerate linear Berge-$C_4$-free $(r+1)$-uniform hypergraphs. Both results are based on the ordered random pick process originated from Martinsson and Steiner. By choosing different selection functions and iteration, we prove the output of the random process gives the required random independent set.

We prove that deciding whether a graph has lettericity at most k is NP-complete, even for four-colorable comparability graphs. Our reduction maps a bipartite graph G to the graph obtained from its incidence graph by inflating each vertex by a clique or an independent set of three vertices. The lettericity of this graph is determined by the numbers of vertices and edges of G and the greatest number of edge-disjoint paths on four vertices in G. Teypaz and Rapine showed that deciding whether the edges of a bipartite graph can be partitioned into paths on four vertices is NP-complete.

Let $q=2^n$ with $n\ge6$, and let $\mathbb{F}_q$ be the finite field with $q$ elements. For a subspace $E$ of $\mathbb{F}_q$, define \[ S(E)=\sum_{x\in E\setminus\{0\}}x^{-1}.\] Let $N_{n,k}$ denote the number of $k$-dimensional $\mathbb{F}_2$-subspaces $E$ for which $S(E)=0$. We prove that, if $k\ge3$ and $n\ge2k+1$, then \[ \left|\frac{2^nN_{n,k}}{\genfrac{[}{]}{0pt}{}{n}{k}_{2}}-1\right|<2^{r^2+r}2^{n(1-r/2)},\] where $r=\lfloor\frac{k-1}{2}\rfloor$, and $\genfrac{[}{]}{0pt}{}{n}{k}_{2}$ is the Gaussian binomial coefficient. In particular, \[ N_{n,k}\sim2^{-n}\genfrac{[}{]}{0pt}{}{n}{k}_{2}\quad (n\to\infty)\] uniformly for $n/3<k\le(n-1)/2$. Combined with the known low-dimensional cases and the symmetry between dimensions $k$ and $n-k$, and the elementary middle-dimensional subfield construction, this proves a conjecture of Carlet: for every $3\le k\le n-3$, there exists a $k$-dimensional $\mathbb{F}_2$-subspace $E$ such that $S(E)=0$.

The classical Descartes' rule of signs bounds the number of positive roots of a real polynomial in one variable by the number of sign changes of its coefficients. This bound only depends on the order of the exponents and on the signs of the coefficients, that is, on the oriented matroids of the exponent and coefficient matrices. We give a multivariate generalization of this rule to arbitrary fewnomial systems: for systems of $n$ real polynomial equations in $n$ variables with $n+k+1$ monomials, we obtain upper bounds for the number of nondegenerate positive solutions which only depend on sign conditions on the maximal minors of the coefficient matrix and of the exponent matrix and its liftings. For $k=1$ they are the Descartes rules of signs for circuits of Bihan-Dickenstein and Bihan-Dickenstein-Forsgård. Our approach starts from the Gale duality and the Khovanskii-Rolle argument of Bihan and Sottile, which reduce the problem to counting common zeros of iterated Jacobians in a polyhedron $\Delta$. This polyhedron is a chamber of a hyperplane arrangement whose combinatorics is determined by the oriented matroid of the coefficient matrix. We bound the number of these zeros by the Bézout number minus the zeros forced in the other chambers by a topological degree, which we compute by a facet-count formula involving the signs of minors of a Gale dual matrix of the exponent matrix. This leads to a recursion over the flats of the arrangement, which ends on lines with the Descartes rules for circuits. In this way the coefficients enter the bounds through the combinatorics of the arrangement, and the exponents through sign conditions on minors.

Motivated by Gal's conjecture, Ferroni and Higashitani conjectured that the $h^*$-polynomial of any Gorenstein lattice polytope admitting a quadratic triangulation is $\gamma$-positive. Together with conjectures predicting quadratic properties of toric ideals of smooth lattice polytopes, this suggests that smooth Gorenstein lattice polytopes should have $\gamma$-positive $h^*$-polynomials. In this paper, we prove that the Ehrhart polynomial of the polar dual of every pseudo-symmetric simplicial reflexive polytope is magic positive. For pseudo-symmetric smooth Fano polytopes, we prove the stronger statement that all magic coefficients are strictly positive. Consequently, for every pseudo-symmetric simplicial reflexive polytope, its polar dual is Ehrhart positive and has a real-rooted and $\gamma$-positive $h^*$-polynomial. We also prove that the Ehrhart polynomial of the polar dual of the symmetric edge polytope of every cycle is magic positive. This gives an affirmative answer to a question of Konoike, who had previously proved partial positivity results for this family.

We study the problem of constructing highly nonlinear vectorial maps $F: \mathbb F_2^n \to \mathbb F_2^m$. Concretely, we want an $F$ and an $A = A(m,n)> 0$ as small as possible, so that for every affine map $L: \mathbb F_2^n \to \mathbb F_2^m$ (of the form $L(x) = M x + b $) we have: $$\mathrm{agree}(F, L) := |\{ x \in \mathbb F_2^n \mid F(x) = L(x) \}| \leq A.$$ Such questions have been studied by Nyberg (1991,1993), Carlet and Ding (2004,2007), Liu, Mesnager and Chen (2017), Nagy (2025), and Biryukov, Turecek, and Udovenko (2026). There is a classical method of constructing such functions from bent-functions and Fourier analytic ideas; the best bound achievable by this method is: $$ A(m,n) = \Theta(2^{n-m} + 2^{n/2}),$$ and in particular, is never smaller than $2^{n/2}$. In this work, we show how to construct highly nonlinear functions beyond this Fourier bound. Concretely, we show how to construct for every $\gamma>0$, a function $F: \mathbb F_2^n \to \mathbb F_2^m$ with $m = O_{\gamma}(n)$, achieving $$ A(m,n) \leq (1 + \gamma)^n.$$ Surprisingly, we even achieve the same quantitative behavior for the much harder question of having low agreement with $m$-tuples of degree $d$ polynomials $Q: \mathbb F_2^n \to \mathbb F_2^m$, with $m = O_{\gamma, d}(n)$. Here the previously best bounds were of the form $A(m,n) = O( 2^{-\frac{n}{2^{d+1}}} \cdot 2^n )$ of Ben-Sasson and Kopparty (2010), based on Gowers-norm-type arguments. All our results generalize to all finite fields $\mathbb F_q$ in place of $\mathbb F_2$. Our methods are based on a new connection to classical results on counting solutions to systems of polynomial equations via algebraic methods. This connection brings us to basic questions in combinatorics, about graphs and hypergraphs with simultaneously a small number of edges and independent sets.

In this paper we construct new examples of diameter graphs and Reuleaux polyhedra in $\mathbb{R}^3$, obtaining a full characterization of their combinatorial structure. For a finite set of points $X\subset\mathbb{R}^d$, its diameter graph is the graph on vertex set $X$ where pairs forming a diameter pair are connected by an edge. Grünbaum, Heppes and Straszewicz independently proved that the diameter graph of $X\subset \mathbb{R}^3$ has at most $2|X|-2$ edges, answering a question of Vázsonyi. Their proof relied on ball polytopes. The ball polytope $\mathcal{B}(X)$ is the intersection of the unit balls centered at the points of $X$. We call a ball polytope a Reuleaux polyhedron if the centers form a family with $2|X|-2$ diameter pairs. Kupitz, Martini and Perles showed that the skeleton of a Reuleaux polyhedron must be a 2-connected strongly involutive self-dual graph. They conjectured that in the simple 3-connected case this is also sufficient. We not only confirm this conjecture, but we show that any 2-connected (not necessarily simple) strongly involutive self-dual graph arises as the skeleton of a Reuleaux polyhedron. To construct the new Reuleaux polyhedra we construct new diameter graphs. It was known that any 3-dimensional diameter graph is a subgraph of a non-bipartite quadrangulation of the projective plane. We show that the reverse holds. That is, for any such graph we construct a diameter realization. This also confirms and strengthens a conjecture of Montejano, Pauli, Raggi, Roldán-Pensado on metric embeddings of strongly involutive self-dual graphs. The construction relies on ideas from rigidity theory. We also discuss a number of applications of these results, such as the construction of bodies of constant width and connections to Steinitz's theorem and Borsuk's conjecture.

This paper investigates the preservation of three classes of Ehrhart properties under various operations on lattice polytopes. These operations include Cartesian products, lattice joins, lattice pyramids, free sums, Minkowski sums, Cayley sums, reflexive polarity, and integral dilations. Specifically, we consider the following properties frequently studied in Ehrhart theory: (i): Positivity of Ehrhart coefficients, including Ehrhart positivity and magic positivity. In particular, we present lattice point counting formulas for the polytopes generated by these operations. (ii): Coefficient properties of the $h^*$-polynomial, including symmetry, unimodality, log-concavity, ultra log-concavity, real-rootedness, and $\gamma$-positivity. (iii): Geometric properties, including the spanning property, the integer decomposition property, very ampleness, and the existence of unimodular triangulation, regular unimodular triangulation, and quadratic triangulation. We determine which properties are preserved under these eight operations, establishing preservation theorems or constructing explicit counterexamples. Furthermore, when a property is not preserved in general, we investigate sufficient or equivalent conditions for its preservation.

In this series of articles, we study two graph classes and their various geometric representations. The first class consists of strongly involutive self-dual graphs (SISD graphs); the second consists of non-bipartite quadrangulations of the projective plane (NBQP graphs). These classes naturally arise in many geometric problems. For example, NBQP graphs are connected to diameter graphs in $\mathbb{R}^3$, tangency graphs of families of pairwise intersecting circles and pseudocircles, generalized thrackles, and graphs formed by the main diagonals of negatively self-polar polytopes. On the other hand, SISD graphs correspond to the skeletons of negatively self-polar polytopes, self-dual cones, and extremal ball-polytopes. In Part I we show a natural one-to-one correspondence between non-bipartite quadrangulations of the projective plane and 2-connected strongly involutive self-dual maps. In particular, we show that the reduced vertex-face incidence structure of a 2-connected strongly involutive self-dual map is a non-bipartite quadrangulation of the projective plane, and every non-bipartite quadrangulation of the projective plane arises that way. This allows us to translate natural properties from one class to the other. These results lay the foundation for later parts of the series.

Let \(\mathbb{F}_q\) be a finite field with \(q>7\). Zhi-Wei Sun conjectured that for every \(a_0\in\mathbb{F}_q\), there is a circular permutation \((a_1,\dots,a_{q-1})\) of non-zero elements of \(\mathbb{F}_q\) such that \(a_0+a_i a_{i+1}\) is primitive for \(1\le i\le q-1\), where \(a_q=a_1\). In this paper, we confirm this conjecture for \(q>18\,888\,871\). For \(a_0=0\), the conclusion holds for all \(q>4\). For \(a_0\ne0\), we apply the Chvátal–Erdős theorem to obtain a Hamilton cycle.

We provide in this text a new framework for a family of combinatorial Hopf algebras. We adopt for this the language of species and twisted bialgebras (also known as Hopf monoids). We define the category of combinatorial twisted bialgebras (briefly, CTBs), a class of twisted bialgebras satisfying strong combinatorial conditions of compatibilities of the product and the coproduct with the underlying combinatorial structure. The considered morphisms also have to respect the combinatorial structure, in some sense. Many classical examples fit into this framework, including Hopf algebras of graphs, rooted trees, posets, hypergraphs, symmetric functions, boolean maps, etc. Most of known combinatorial Hopf algebras can be obtained from a CTB by application of a Fock functor, or by duality. A central result is the introduction of a universal CTB denoted by $\mathbf{Exch}$, built from collections of compositions satisfying a certain exchange condition. It has a special character $\epsilon$, such that the pair $(\mathbf{Exch},\epsilon)$ plays a role analogous to that of quasi-symmetric functions for graded and connected Hopf algebras: it is the terminal object in the category of CTBs equipped with combinatorial characters. This leads to universal morphisms that encode important information about the combinatorial structure of coproducts. The natural order on the set of compositions on a fixed set is used to define three important sub-objects $\mathbf{Exch}\_-$, $\mathbf{Exch}\_+$ and $\mathbf{Exch}\_{+\hspace{-2mm}+}$ of $\mathbf{Exch}$. We also studies polynomial invariants associated with CTBs. It establishes a correspondence between characters and polynomial invariants. Well-known invariants such as the chromatic polynomial of graphs and the Ehrhart polynomials of posets arise naturally within this framework. We show that, for a given degree, only finitely many polynomial invariants can occur. We show that these universal polynomial invariants can be used to give cancellation-free formulas for the antipode or the Eulerian idempotent, under conditions on the CTB, and use this for an example based on rooted trees, coming from a Hopf algebra used in linguistics. Results are also obtained for families of polynomial invariants with the alternating sign condition, or satisfying a reciprocity principle. We also extend the theory to combinatorial twisted double bialgebras (briefly, CTDBs), which involve two interacting coproduct structures. These generalize previously known examples arising from graphs, posets, hypergraphs, and rooted trees. The twisted approach preserves the combinatorial richness of these structures while providing a unified algebraic framework. We show that not any CTB can be made a CTDB, leading to the introduction of the notion of quasi-cocommutativity. In particular, we show that $\mathbf{Exch}\_+$ is not a CTDB, and construct a sub-object which is, in a unique way, and maximal for the inclusion. In the cocommutative case, simplifications naturally occur, and we show that in this case, a CTB of hypergraphs plays a central role, especially two sub-objects based on connectedness systems and antichain covers. We also show that, in the cocommutative case, the universal polynomial invariants are always chromatic polynomials of hypergraphs, no matter which CTB is considered. Finally, forgetting the combinatorial underlying structure, we study the algebraic structure of $\mathrm{Exch}$, with the help of a twisted version of Loday and Ronco's rigidity theorem for infinitesimal bialgebras.

A unit-distance graph is a graph which admits a realisation in Euclidean space in which every edge has unit length. Imposing further geometric conditions on the non-edges gives a family of natural subclasses. Requiring that no two vertices lie at distance less than one gives penny and marble graphs: the contact graphs of collections of equal radii $d$-dimensional spheres with non-overlapping interiors for $d=2,3$. Requiring instead that the straight-line drawing in the plane be non-crossing gives matchstick graphs. Since any motion of a realisation must preserve these extra conditions, the rigidity and flexibility properties of the resulting frameworks differ from those of classical bar-joint rigidity theory. In this note we analyse various rigidity problems for penny and marble graphs, matchstick graphs and unit-distance graphs. In particular we answer a recent open problem on penny and marble graph rigidity and establish a link between penny graphs and the concept of NAC-colourings.

We obtain improved Fourier restriction and Kakeya maximal estimates in $\mathbb R^4$. More precisely, by proving the two-ends incidence estimates $\mathbf{TE}\left(\frac{133}{44},\frac{177}{88},\frac34\right)$ and $\mathbf{TE}\left(\frac{386453}{125988},\frac{386453}{125988},\frac{358951}{377964}\right)$, we extend the restriction range to $p>2+ \frac{176}{221}$ and the Kakeya maximal estimate up to dimension $3.06737$. The implications to the known lower-bounds for the Hausdorff dimension of Kakeya sets in $\mathbb{R}^4$ and the Bochner–Riesz conjecture are also studied.

Let $w \colon (0,1) \to \mathbb{R}_{+}$ be a weight. We prove that for an arbitrary Schauder basis $\{e^{i \lambda_n t}\}_{n \in \mathbb{Z}}$ in $L^2(0,1)$ and an arbitrary lacunary defect set $A \subset \mathbb{Z}$ the system $\{w(t) r_n(t)\}_{n \in \mathbb{Z} \setminus A}$ is always complete and never minimal for any weight $w$ satisfying a natural decay condition, which is sharp on the exponential scale. Moreover, we establish a simple combinatorial criterion for systems of the form $ \{ w(t) e^{2\pi i n t} \}_{n \in \mathbb{Z} \setminus A} $ to be complete and minimal in $L^2(0,1)$ for an arbitrary weight $w$ and a finite defect set $A \subset \mathbb{Z}$.

We investigate derivation modules of hyperplane arrangements associated with the monomial groups $G(r,p,\ell)$. Motivated by Mühlherr's separator-based construction for graphic arrangements, we extend this approach by combining separators with integral expressions. For a subarrangement $\mathcal{A}_G$ defined by a graph $G$, we show that, when $p<r$, the derivation module $\mathcal{D}(\mathcal{A}_G)$ is generated by the standard derivations $\eta_0,\dots,\eta_{\kappa(G)}$ together with a finite set of separator-based derivations. This result generalizes the complete graph case and provides new evidence toward the broader applicability of separator methods in the study of hyperplane arrangements.

We construct explicit off-diagonal Ramsey graphs from polynomial families of incident point–hyperplane flags. Universal interpolation translates the exact dimension of ordered clique configurations into a bound on their coefficient–label incidence. At the critical dimension, a polynomial separates the forbidden-pair image from its diagonal; deleting the corresponding edges preserves a triangular rank certificate. Restriction of scalars realizes the rational endpoint without rounding loss. The resulting fixed-$s$ exponent has leading scale $s/(2\log_2s)$, and short Frobenius relations give the explicit example $R(16,t)\ge\Omega(t^{2.0539221767\ldots})$. A finite-fiber construction gives the range $s^2(\log s)^2=o(\log t)$. For fixed parameters, algebraic preprocessing terminates and field initialization, vertex decoding and adjacency take deterministic time polynomial in the extension degree. We also retain an elimination-free variant and formulate the filtering argument for general forbidden configurations with a distinguished edge.

Let $Y_{n,r,q}$ be the graph obtained from the Turán graph $T_{n,r}$ by adding $q$ pairwise disjoint edges inside a largest part, and let $c(n,F)$ be the minimum number of copies of $F$ created by adding a single edge to $T_{n,r}$. Fang, Li, Lin and Ma proved that for every color-critical graph $F$ with $\chi (F)=r+1$, there exists a constant $\delta_F>0$ such that for all sufficiently large $n$ and all $1\le q\le \delta_F \sqrt{n}$, the condition $\lambda(G)\ge\lambda(Y_{n,r,q})$ forces at least $q\, c(n,F)$ copies of $F$. The bound $q=O(\sqrt{n}\,)$ is tight up to a constant factor, in contrast to the linear order $n$ of the edge setting of Mubayi, Pikhurko and Yilma, but the exact constant $\delta_F$ remained unknown for any $F$. In this paper, building on a structural result of Fang, Li, Lin and Ma, we determine the threshold $\delta_F$ when $F$ is a clique and an odd cycle. For every $r\ge2$, we denote $\delta_r :=(1-\tfrac1r)\sqrt2$ and prove that for every $\varepsilon>0$, if $n$ is sufficiently large and $1\le q\le(\delta_r-\varepsilon)\sqrt n$, then every $n$-vertex graph $G$ with $\lambda(G)\ge\lambda(Y_{n,r,q})$ contains at least $q\,c(n,K_{r+1})$ copies of $K_{r+1}$, and $\delta_r$ is best possible. For odd cycles, the threshold is $1/\sqrt2$. For every $k\ge1$ and $\varepsilon>0$, if $n$ is sufficiently large and $1\le q\le(1/\sqrt2-\varepsilon)\sqrt n$, then every $n$-vertex graph $G$ with $\lambda(G)\ge\lambda(Y_{n,2,q})$ contains at least $q\,c(n,C_{2k+1})$ copies of $C_{2k+1}$, and $1/\sqrt2$ is best possible. Our results determine both the exact count of copies and the optimal range of $q$. The behavior in the spectral setting differs from the classical edge setting, in which the range of $q$ is of order $n$ and the threshold is $1/r$ for cliques by Lovász and Simonovits, and $1/2$ for odd cycles by Pikhurko and Yilma.

Oxley conjectured that a matroid with a circuit-cocircuit intersection of size $k\geq 4$ has one of size $k-2$. Using the standard polynomial representation of the classical extended binary Golay code, we construct an identically self-dual binary matroid of rank $12$ on $24$ elements. Its circuit-cocircuit intersection sizes are exactly $0,2,4,6,8$, and $12$, disproving the conjecture for $k=12$. The proof is self-contained and does not rely on the full weight distribution of the code or on an enumeration of all circuits.

Let $A$ be the (non-symmetric) adjacency matrix of a uniformly random $d$-regular directed graph on $n$ vertices, and let $\Xi$ be independent of $A$ with i.i.d. Rademacher entries. Suppose that $\min(d,n-d)\ge\lambda n$ for some fixed $\lambda\in(0,1/2]$. We show that there exists $c > 0$, depending only on $\lambda$, such that \[ \mathbf P_{A,\Xi}\{\operatorname{rank}(A\circ\Xi)\le n-k\}\le e^{-c nk},\qquad 1\le k\le n. \] As an ingredient in the proof of the rank bound, we use the case $\kappa=0$ of the following quantitative smallest singular value estimate: \[ \mathbf P_{A,\Xi}\{s_n(A\circ\Xi)\le\kappa\} \le C\kappa\sqrt n+e^{-c' n}, \qquad \kappa\ge0, \] where $C,c'>0$ depend only on $\lambda$.

A $k$-bend path is a non-self-intersecting polyline that lies on a grid and consists of at most $k+1$ axis-parallel line segments. A $B_k$-CPG graph is a graph whose vertices can be represented by pairwise interiorly disjoint $k$-bend paths on a grid such that two vertices are adjacent if and only if the corresponding grid paths touch at a grid point. We prove that recognizing planar $B_0$-CPG graphs of maximum degree 8 is NP-complete, and that recognizing planar $B_1$-CPG graphs of maximum degree 11 is NP-complete. These results settle two of the three planar recognition problems left open by Champseix, Galby, Munaro, and Ries.

Let $\varphi$ be the golden ratio and let $R_n=\{\lfloor k\varphi^n\rfloor : k\ge 1\}$ be the $n$-th row of the array $T(n,k)=\lfloor k\varphi^n\rfloor$ (OEIS A128440). In 2022 Kimberling conjectured that the rows $R_{2n-1}$ and $R_{2n}$ are disjoint, and that after the two rows are merged and each entry is replaced by its rank, they become the lower and upper Wythoff sequences. He also conjectured (OEIS A358359) that if $a(N)$ is the number of rows containing $N$, then every positive integer occurs infinitely often among the values of $a$. We show that the first two conjectures follow quickly from the Skolem-Bang theorem, which also yields the exact rule for when two rows are disjoint: $R_i\cap R_j=\emptyset$ ($i<j$) if and only if $j-i$ is odd and divides $i$. We then prove the third conjecture. The main tools are an explicit determination of the rows containing an odd-indexed Lucas number, which extends a result of Noppakaew, Kanwarunyu and Wanitchatchawan, and a "Lucas shift" lemma: if $N+1$ is not of the form $L_{2e}$ with $e\ge1$, then adding a sufficiently large even-indexed Lucas number to $N$ does not change the set of rows containing it. We also show that each value of $a$ is taken on a set of positive natural density, and we report computations up to $10^8$ suggesting that the least $N$ lying in exactly $v\ge 2$ rows is the Lucas number $L_{4v-5}$.

We construct arbitrarily large finite sets $A$ of real algebraic integers such that both $|A+A|$ and $|AA|$ are at most $|A|^{1.95835}$. The construction combines truncated ideal-smooth $S$-unit fibres with a coprime additive factor. A tensor-product rank argument, using a two-dimensional local feature at each selected prime ideal, controls the loss in the additive factor, while a direct estimate for an outer parallel body improves the sumset packing bound. The arithmetic input is an unramified pro-$2$ tower over a known degree-ten field, with simultaneous Frobenius cuts controlling the small prime ideals. We use unconditional Tsfasman–Vlăduţ inequalities for the joint class-number–regulator cost. All finite numerical comparisons entering the exponent are certified by outward rational interval arithmetic. No unproved hypothesis is used.

Let $G$ be a simple graph on $n$ vertices, and let $F(G)$ denote the number of its spanning forests. Bencs and Csikvári [Upper bound for the number of spanning forests of regular graphs, European J. Combin. 110 (2023) 103677] proved that every $r$-regular graph $G$ with $r\geq 2$ satisfies $F(G) \leq r^{n}$. They further conjectured that for $r \geq 3$, \[ F(G)^{1/n} \leq \frac{(r - 1)^{r-1}}{(r^2 - 2r - 1)^{r/2-1}}. \] In this paper, we resolve this conjecture in the affirmative.

We study $W$-free planar graphs for $W\in\{W_{2,4},W_{2,5},W_{3,4}\}$, where the quasi-double star $W_{h,k}$ is obtained from a three-vertex path by attaching $h$ leaves to one endpoint and $k$ leaves to the other. We prove that every $W_{2,4}$-free planar graph on $n$ vertices has at most $9n/4$ edges, and the bound is attained whenever $8\mid n$. This determines the planar Turán density of $W_{2,4}$ as $9/4$. We also establish the sharp upper bound $5n/2$ for $W_{2,5}$. Combined with known constructions of planar graphs of maximum degree five, it yields $\ex_{\PP}(n,W_{2,5})=\lfloor5n/2\rfloor$ for every $n\ge15$. These results close the two corresponding coefficient gaps in the bounds of Liu et~al. Our proofs use structural restrictions on high-degree vertices, local deletions, and degree deficits in neighborhoods of radius two. For $W_{3,4}$, we characterize the planar graphs with a dominating vertex that avoid this tree and determine their exact extremal number, $\lfloor(5n-7)/2\rfloor$, for every $n\ge10$. Finally, $W$-free planar triangulations have at most eight, twelve, and eleven vertices for $W=W_{2,4},W_{2,5},W_{3,4}$, respectively; the first two bounds are sharp.

Partly motivated by enumeration of parking functions and their variants, there is a long-standing interest in lattice paths refined by several statistics, including the notable $\mathsf{run}$ and $\mathsf{return}$. We consider bounded lattice paths, which are bounded by a given lattice path and are related to vector parking functions, for which we generalize $\mathsf{run}$ to composition runs, parameterized by a composition. By constructing involutions on these paths, we establish symmetries relating composition runs to some generalized return statistics. As an application, we settle an open problem of Dai, Fu, and Qiu on rational Dyck paths. The symmetries between generalized $\mathsf{run}$ and $\mathsf{return}$ are then transferred to vector parking functions, which suggest a new notion of prime decomposition. In the special case of $(a, b)$-parking functions, we compute the generating function refined by $\mathsf{run}$ and $\mathsf{pri}$, which is new even for classical parking functions.

We study the hitting time index, a graph invariant defined in terms of expected hitting times of simple random walks on finite connected graphs, through its relation with the Kirchhoff index. Using the commute time identity and Tetali's formula, we decompose the hitting time index into a Kirchhoff-index term and a nonnegative asymmetry term expressed by degree-weighted effective resistance sums. We derive lower and upper bounds for the asymmetry term, together with their equality conditions, and obtain a lower bound for the hitting time index in terms of the range of the degree-weighted resistance sums. For trees, we obtain lower and upper bounds for the asymmetry term, with the upper bound expressed in terms of the Wiener index, and characterize the equality case in the lower bound. In particular, we show that among all trees of fixed order, the hitting time index is minimized uniquely by the star graph. Finally, we apply the decomposition to complete bipartite graphs, paths, and graphs obtained by conjoining complete graphs, and recover several known hitting-time formulas.

Tutte's flow conjectures ask when a graph has a nowhere-zero flow with small integer values. We approach these questions through spanning-tree packing and an analytic minimization argument of Alon, Bucić, and Davies. For every integer $q\ge2$, we prove that $q$ edge-disjoint spanning trees suffice to realize every compatible outdegree prescription modulo $q$. For odd prime moduli this proves the graphic Additive Base Conjecture. More generally, integer intervals of edge values realize every prescribed boundary when their widths satisfy the corresponding tree-packing inequalities. We strengthen this condition by allowing a deficit of two in every partition inequality, provided every cut has total width at least $q-1$. The allowance of two is sharp for $q\ge3$. Consequently, five-edge-connected graphs with at most ten odd-degree vertices are $\mathbb Z_3$-connected, and three-edge-connected graphs with at most fourteen odd-degree vertices are $\mathbb Z_5$-connected. Every orientation of a nine-edge-connected graph with at most fourteen odd-degree vertices admits an antisymmetric $\mathbb Z_5$-flow. The refinement also gives sharp bounds for extending preorientations after deleting edges or vertices, including the boundary case of $2q$-edge-connectivity. The same argument recovers Seymour's six-flow theorem, the three-flow theorem for six-edge-connected graphs, and the four-flow theorem for graphs with two edge-disjoint spanning trees. It also gives sharp circular-flow bounds and a cycle-rank bound strictly below six. We examine a five-flow reformulation obtained by tripling the edges of cubic graphs. A $46$-vertex graph of fractional packing number $41/9$ with no modulo-five orientation shows that packing greater than $9/2$ alone does not guarantee such an orientation in arbitrary graphs.

For a fixed integer $k\ge 2$, let $kK_3$ denote the vertex-disjoint union of $k$ triangles. A recent fixed-size spectral theorem of Das and Yamini asserts that, for all sufficiently large $m$, every $kK_3$-free graph $G$ of size $m$ satisfies $\lambda(G)\le (k-1)+\sqrt{m-k(k-1)},$ and equality holds if and only if $(2k-1)\mid m$ and $ G\cong \bigl(K_{2k-1}\vee qK_1\bigr)\cup tK_1,\, q=\frac{m}{2k-1}-(k-1) $ for some $t\ge 0$. They explicitly posed the open problem: Let $k\ge 2$ be fixed and $\ell$ be a residue in $\{1,\dots,2k-2\}$. For all sufficiently large integers $m\equiv \ell\pmod{2k-1}$, determine the exact value of $\max\bigl\{\lambda(G): e(G)=m,\ G\text{ is }kK_3\text{-free}\bigr\}, $ and characterize all graphs attaining this maximum. In this paper, using the positive-defect version of the bounded-core method for divisible sizes together with several new ideas developed in this paper, we give a complete solution to the aforementioned open problem.

For every $N\ge36$, we realise any prescribed monic irreducible binary polynomial of degree $N$ as the characteristic polynomial of a linear hybrid cellular automaton on a near-square region of exactly $N$ cells. The transition matrix is Moore-local with a null boundary and has at most $3N-1$ directed nonself dependencies. The dependency graph retains a bidirectional Hamilton path; its indegree, outdegree and underlying undirected degree are at most six. The underlying graph contains an explicit square grid minor of side proportional to $\sqrt N$. The deterministic synthesis takes $O(N^3)$ bit operations. The construction combines a local similarity transformation with a transport potential and joint routing across consecutive row gaps. Exact verification of finite certificates, followed by induction, proves the required routing inequalities for every admissible width. We also give an entirely analytic construction with fewer than $7N/2$ dependencies and prove a lower bound of $5N/2-O(\sqrt N)$ for the retained path and full rectangular grid minor. This lower bound is sharp when the characteristic polynomial is unrestricted.

Every balanced instance of the stable marriage problem with strict complete preferences has a unique finest partition into prime blocks, and that single partition simultaneously factors three different structures: the reachable execution digraph as a Cartesian product, the proposal-prefix antimatroid as a direct sum, and the stable-matching lattice as a direct product. The converse fails, and fails at every size from two on: two explicit families share the identical Boolean-cube execution while one is maximally decomposable with a single stable matching and the other is prime with n. Uniqueness yields an exact census, a recursion counting the prime instances at every size, under which exactly 88,478,208 of the 110,075,314,176 profiles with four agents on each side are decomposable and the decomposable fraction is asymptotically n! / n^(2n). The blocks are characterised as the square components of the mutual-rank filtration, so the partition is computable in polynomial time and the factorisation is a tool rather than only a fact.