Papers

Given an $n\times n$ matrix $A=(a_{i,j})$ and subsets $I,J$ of $[1,n]:=\{1,\dotsc,n\}$, let $A_{I,J}=(a_{i,j})_{i\in I,j\in J}$ denote the $(I,J)$-submatrix of $A$. We prove that, for every totally nonnegative matrix $A$ and every $h=1,\dotsc,n-1$, \begin{equation} \operatorname{perm}(A_{[1,h],[1,h]}) \operatorname{perm}(A_{[h+1,n],[h+1,n]}) \;\geq\; \operatorname{perm}(A_{[1,n]\cap 2\mathbb Z,[1,n]\cap 2\mathbb Z}) \operatorname{perm}(A_{[1,n]\smallsetminus 2\mathbb Z,[1,n]\smallsetminus 2\mathbb Z}). \label{eq:absmain} \end{equation} In particular, these inequalities hold for the $0$-$1$ totally nonnegative matrices arising as antiadjacency matrices of unit interval orders. Our proof is based on explicit Bruhat-order comparisons between two distinguished families of permutations. Let $\mathfrak{A}_n$ be the Young subgroup $\mathfrak{S}_{1,\ldots,\left\lfloor\frac{n}{2}\right\rfloor} \times \mathfrak{S}_{\left\lfloor\frac{n}{2}\right\rfloor+1,\ldots,n}$ and let $\mathfrak{B}_n$ be the set of parity-alternating permutations \begin{equation*} \{w=w_1\cdots w_n\in\mathfrak{S}_n \mid i \text{ and } w_i \text{ have the same parity}\}. \end{equation*} For the balanced case $h=\left\lfloor\frac{n}{2}\right\rfloor$, we construct a bijection $f_n:\mathfrak{A}_n\rightarrow\mathfrak{B}_n$ satisfying $w\leq f_n(w)$ in Bruhat order. We then extend this construction to arbitrary $h$ by means of maps $g_{n,h}$ satisfying the corresponding Bruhat-order comparison. These maps yield the inequalities for all totally nonnegative matrices.

We prove ten conjectures of Z.-W. Sun on Ramanujan-type series for $1/\pi^4$ and $\pi^4$ with harmonic numbers. They follow by comparing Taylor coefficients in new Borwein-Bradley-type identities, whose products depend only on $x^3$, $x^4$, $x^6$ or $x^8$. These identities come from parametric extensions of the series of Cullen and Zhao. We derive the extensions from K. C. Au's Wilf-Zeilberger (WZ) seeds, and their sums are trigonometric. Our proofs use Guillera's periodicity argument and give new proofs of the original series. In the same way, Dougall's $_5F_4$ sum gives half-integer analogues of the identities of Koecher, Borwein-Bradley and Cohen, whose Taylor coefficients are Apéry-like series for products of zeta values.

In this paper we study domination and total domination of Johnson graphs $J(n,k)$. We establish monotonicity results proving that both $\gamma(J(n,k))$ and $\gamma_t(J(n,k))$ are non-decreasing with respect to $n$. We compute exact values for $\gamma_t(J(n,k))$ in specific cases, showing that $\gamma_t(J(n,2))=\lceil\frac{2}{3}(n-1)\rceil$ for $n\geq 4$, and determining $\gamma_t(J(n,3))$ for $n\geq 6$, which depends quadratically on $n$.

The oriented diameter of a connected bridgeless graph is the minimum diameter of a strong orientation. Let $f(d)$ be the maximum oriented diameter among all such graphs of diameter $d$. In 1978, Chvátal and Thomassen proved $f(d)\le2d^2+2d$ and constructed graphs showing that any quadratic upper bound on $f(d)$ must have leading coefficient at least $1/2$. We prove that $f(d)\le \tfrac12d^2+7d$ for every integer $d\ge1$. This matches the leading coefficient of their lower bound and establishes $f(d)=\tfrac12d^2+O(d)$, determining the optimal quadratic coefficient. Our proof gives a polynomial-time algorithm that constructs a strong orientation satisfying the stated bound.

For a finite graph $G$, the maximum average degree $\operatorname{mad}(G)$ is the largest average degree of a nonempty subgraph of $G$. Hendrey, Norin and Wood asked whether this parameter is partitionable, that is, whether for all positive reals $a,b$ every graph $G$ with $\operatorname{mad}(G)<a+b$ admits a vertex partition $V(G)=A\cup B$ with $\operatorname{mad}(G[A])<a$ and $\operatorname{mad}(G[B])<b$. In this note, we answer this question affirmatively. Moreover, we generalize this to an arbitrary number of partition classes. The clustered chromatic number $\chi_\star(\mathrm{G})$ of a graph class $\mathrm{G}$ is the minimum integer $k$ such that, for some integer $c$, every graph in $\mathrm{G}$ has a $k$-coloring in which every monochromatic component has at most $c$ vertices. We apply this result to show that $\chi_\star(\mathrm{A}_m)=\left\lfloor\frac{m}{2}\right\rfloor+1$, where $\mathrm{A}_m$ is the family of graphs $G$ with $\operatorname{mad}(G)\leq m$. This solves an open problem posed in Wood's survey and highlighted by Hendrey and Wood. As another application of our general partition result for $\operatorname{mad}$, we obtain a bound for relaxed colorings. Namely, for non-negative integers $d_1,\ldots,d_k$, every graph $G$ with \[\operatorname{mad}(G)<\sum_{i=1}^k \frac{2d_i+2}{d_i+2}\] admits a partition $V(G)=V_1\cup\ldots\cup V_k$ such that $\Delta(G[V_i])\leq d_i$ for each $i\in[k]$. In particular, this provides the first non-trivial bounds for $k\ge3$ with arbitrary $d_i$ and improves the previously known bound for $(d+1,d)$-colorings with $d \ge 2$.

We study how the ranks of factors in a positive semidefinite slack factorization constrain its size. We bound the number of low-rank vertex factors in any fixed factorization, and use this to determine the asymptotic order of the minimum factorization size under any fixed bound on vertex-factor ranks: $\Theta(n\log n)$ for permutahedra $\Pi_n$ and $\Theta(\log N)$ for regular $N$-gons. In the rank-one case this gives the order of the minimum dimension of a space of real functions on the vertices in which every facet slack is a sum of squares, with no symmetry or degree restriction. For unrestricted lifts, the bound gives $\mathrm{xc}_{\mathrm{PSD}}(\Pi_n)\ge n+\log_3 n-O(1)$, and shows that a permutahedron factorization of linear size would need vertex factors of rank $\Omega(\log n)$ at all but a vanishing fraction of vertices. We also prove that every polytope of real positive semidefinite rank at most four has at most twelve vertices, and construct explicit size-four lifts for every square-symmetric octagon, so eight is attained. The upper bound combines a count of the polygon corners reached by curves of rank-one factors with incidence constraints on factor ranks. Whether eight is the maximum remains open.

The second-order Zarankiewicz number $z_2(m,n)$ and the biquadratic sum-of-squares rank $\mathrm{BSR}(m,n)$ are related by the unconditional hierarchy \[ \mathrm{BSR}(m,n)\ \ge\ z_2(m,n)\ \ge\ z_{SL}(m,n)\ \ge\ z_{RL}(m,n) \ \ge\ z_{wL}(m,n)\ \ge\ z(m,n). \] We introduce the second-order augmented Zarankiewicz number $z_{2A}(m,n)$, obtained from $z_2(m,n)$ by deleting the requirement that the configuration be limited, so that \[ \mathrm{BSR}(m,n)\ \ge\ z_{2A}(m,n)\ \ge\ z_2(m,n). \] Although the defining class is enlarged, $z_{2A}$ still obeys the universal cell bound of Löfberg and Qi, because that bound uses only the \(C_4\)-freeness of the one-edge graph. We prove \[ \mathrm{BSR}(4,4)\ \ge\ z_{2A}(4,4)\ =\ 11\ >\ 10\ =\ z_2(4,4) \ =\ z_{RL}(4,4), \] the first recorded separation between the second-order number and its augmented variant. This result also gives a better lower bound for $\mathrm{BSR}(4,4)$. The lower bound is witnessed by an explicit non-limited $4\times4$ configuration of displayed length $11$ whose recursive-line closure satisfies $(\mathrm{RW}3^+)$; the matching upper bound excludes $12$ by the universal cell bound together with an exact finite classification of the $161$ twelve-square configurations, and \(z_2(4,4)=z_{RL}(4,4)=10\) is the exact value of Xu and Yan.

We study bootstrap percolation with independent vertex thresholds taking values one and two, with threshold-one probability $(1-c/d)/d$ for fixed $c>0$. The seed set is chosen uniformly among sets of a prescribed deterministic size, independently of the graph and thresholds. We prove threshold statements at fixed relative margins. For uniform simple graphs with prescribed nonnegative integer degrees of even sum and exact mean $d$, assume $\max_i|d_i-d|\le C\sqrt d$ and $c+1-v_n\ge\kappa>0$, where $v_n=\operatorname{Var}(D)/d$ and $C,\kappa$ are fixed. When $d\to\infty$ and $d=o(n^{1/7})$, the leading seed scale is $n(c+1-v_n)^2/(2d^4)$, without requiring a limit of $v_n$. At fixed relative margins below and above this scale, the final active set has size $O(n/d^3)$ and $n-o(n)$, respectively, with high probability. A separate result for $G(n,d/n)$ holds when $d\to\infty$ and $d^5/n\to0$, and gives scale $nc^2/(2d^4)$. Thus regular and independent-edge graphs have different coefficients at the same asymptotic mean degree. Local exploration estimates yield explicit inactive remainders and survive conditioning on simplicity in the prescribed model. We also quantify the precision obstruction to static inclusion transfer and compute deterministic response-barrier corrections, without identifying a shrinking random critical window.

The main aim of this article is to obtain explicit formulas for the root multiplicities of a class of finite-rank symmetrizable Borcherds-Kac-Moody algebras g(A) whose real part is of type A. The Dynkin diagram of each such algebra is obtained from the disjoint union of a Dynkin diagram of type A and a finite simple graph G_im, whose vertices correspond to imaginary simple roots, by choosing one vertex in each and joining the chosen vertices by a single edge of multiplicity c>0. More generally, we consider the same construction with the type A diagram replaced by the Dynkin diagram of an arbitrary finite-rank symmetrizable Kac-Moody algebra g_re. The resulting symmetrizable Borcherds-Kac-Moody algebra g(A) has real part g_re and imaginary part represented by G_im. Our starting point is a graph-theoretic description of the quotient of the denominator products of g(A) and g_re. This quotient is a multivariate independence polynomial of G_im, with the variable corresponding to the attachment vertex weighted by normalized highest-weight characters of g_re. We prove that the coefficients of powers of this quotient factor into a generalized chromatic polynomial and a tensor-power weight multiplicity. Using this, we obtain explicit formulas for all root multiplicities with nonzero imaginary component in terms of generalized chromatic polynomials of G_im and weight multiplicities in tensor products of highest-weight modules of g_re. When G_im is chordal, these formulas specialize to finite divisor sums involving products of binomial coefficients; paths and complete graphs provide particular examples. In particular, we obtain explicit formulas for the root multiplicities of g(A) when its real part g_re is of type A.

We determine the torsion subgroup of $H_{m-1}(\mathbb{B}_mK_{m+1,m+r-1};\mathbb{Z})$ for $m\ge2$ and $r\ge0$: top homology with arbitrary coefficients is the kernel of an unsigned subset-inclusion matrix, and its integral diagonal form determines all primary summands. Every finite order occurs, with explicit representatives. Generalized theta classes span an embedded copy of the cokernel of the inclusion matrix, containing all torsion; for $r\ge m$ they generate the torsion, each of order $\operatorname{lcm}(1,\ldots,m)$. For every prime power $q$ and $m\ge q$, the graph $K_{m+1,m+q-1}$ is minimal in the minor order for order-$q$ torsion in $H_{m-1}(\mathbb{B}_m)$. In particular, odd torsion first appears in $H_2(\mathbb{B}_3K_{4,5})\cong\mathbb{Z}^{155}\oplus(\mathbb{Z}/2)^4\oplus\mathbb{Z}/3$, and no proper minor of $K_{4,5}$ has odd torsion in $H_2(\mathbb{B}_3)$. For arbitrary part sizes, we give a multiplicity-free decomposition of $H_m(\mathbb{B}_mK_{a,b};\mathbb{Q})$ under vertex permutations and prove that $H_{m-1}(\mathbb{B}_mK_{a,b};\mathbb{Z})$ has no $p$-primary torsion when $a,b\ge2m-1$ and $p\ge m$ is an odd prime. The explicit order-$q$ class retains its order under every enlargement of the second part of the graph, while for $m=q=p$ an odd prime it is killed by a specified enlargement of the first part.

A family of $k$-subspaces of an $n$-dimensional vector space over a finite field is $t$-intersecting if any two members intersect in a subspace of dimension at least $t$. A $t$-intersecting family is nontrivial if no $t$-subspace is contained in all its members. In this paper, we determine the largest nontrivial $t$-intersecting families for $k\geq t+2$ and $n\geq 2k+1$. As a result, the complete nontrivial-intersection theorem for vector spaces is established.

We study the largest size $C(n)$ of a subset of the $n\times n$ integer grid containing no isosceles triple, including equally spaced collinear triples. We prove $C(n)=\Omega\left(n\sqrt{\log\log n/\log n}\right)$ by applying the sparse hypergraph coloring theorem of Cooper and Mubayi. Elementary geometry and primitive-direction counts give maximum degree $O(n^2\log n)$ and pair-codegree at most $5n$ for the forbidden-triple hypergraph. The resulting bound improves the explicit guarantee in PatternBoost by a factor of order $\sqrt{\log\log n}$. Recovered interaction records document how a Codex literature proposal, human route selection, finite verification, and author-relayed review informed proof synthesis and repair. We connect these actions to successive proof artifacts and extract two case-derived checking practices: tracking objects, parameters and conclusions in theorem applications, and seeding a definitional omission to test a finite verifier. Separately implemented enumerators agree on complete edge sets for every $2\le n\le12$; suppressing the degenerate-case branch loses exactly the equally spaced collinear triples. These records document assistance under human direction, and finite checks support implementation consistency rather than the asymptotic theorem itself.

Using a dioid algebraic structure, we show that there exist spherically symmetric trees $T(2^m 1^n)$ whose independence polynomials exhibit multiple breaks in log-concavity, a result established by estimating the asymptotic growth of the coefficients of these polynomials. Provided the parameter $n$ is a sufficiently large odd integer, the number of breaks is bounded below by the Jacobsthal numbers. This result affirmatively answers a question raised by D. Galvin [D. Galvin, Trees with non log-concave independent set sequences, arXiv 2502.10654.v2].

Let $P\subset\mathbb R^d$ be finite with $\operatorname{diam}(P)\le1$. For $0<\varepsilon<1$, let $\mathcal N_\varepsilon(P)$ denote the number of ordered pairs at distance at most $\varepsilon$, including the diagonal, and let $\mathcal A_\varepsilon(P)$ denote the number of ordered pairs at distance at least $1-\varepsilon$. We prove the sharp bound \[ \mathcal N_\varepsilon(P)\gtrsim_d \varepsilon^{(d-1)/2}\mathcal A_\varepsilon(P) \] conjectured by Steinerberger (2025). More generally, the same inequality holds for finite positive Borel measures. The main input is a uniform $L^2$ estimate, at cap thickness comparable to lattice spacing, for a supporting-direction incidence operator. In gnomonic coordinates, this operator is controlled by an incidence estimate for small Fenchel–Young gaps.

In this paper we prove the weak Hurwitz stability conjecture for the strong Turán expressions of Eulerian polynomials, as well as the corresponding conjecture for Narayana polynomials, which were conjectured by Liu and Yan. We generalize the notion of Turán expressions of polynomial sequences, and provide a criterion for the generalized stability of generalized Turán expressions. Based on our sufficient condition, we not only confirm Liu and Yan's conjectures but also obtain the generalized stability of generalized Turán expressions of a class of recurrent polynomial sequences and generalized Narayana polynomials.

We study the proper hat guessing game on graphs, introduced by Adriaensen et al. in Hat guessing with proper colorings. In this game, the players are seated on the vertices of a graph $G$ and assigned hats from a set of $k$ colors such that the resulting assignment forms a proper coloring. The visibility of each vertex is limited to the hat colors of their neighborhood. Then they must simultaneously output a guess about the color of their own hat. The players win if at least one guess is correct. A parameter related to this problem is the proper hat guessing number $\operatorname{HG}_{P}(G)$ that is the maximum number of colors $m$ such that the players can guarantee a winning strategy. Motivated by the work of Shurman et al. \cite{shurman2026upper}, we establish the first upper bound that depends both on the number of vertices $n$ and the maximum degree $\Delta$ in the case where $\Delta \geq \frac{n}{e+1}$. This result leads us to show that the proper hat guessing number of the binomial random graph $G_{n,1/2}$ is bounded above by $cn$, where $c \approx 1.366$. Finally, we prove that graphs of maximum degree $(1-\gamma)n$ for some fixed $\gamma \in (0,1]$ cannot have $\operatorname{HG}_{P}(G) = (2-o(1))n$.

Regular semisimple Hessenberg varieties are subvarieties of flag varieties defined for any reductive group. Their cohomology carries a representation of the associated Weyl group, known as the dot action representation. In type $A$, these representations can be computed using the chromatic quasisymmetric functions of naturally labeled unit interval graphs. We prove that restricting the dot action representation of any Weyl group to a type $A$ parabolic subgroup yields a representation isomorphic to a direct sum of type $A$ dot action representations. This work is motivated by the conjecture that an analogous property holds for all parabolic subgroups. In particular, for classical Weyl groups of types $B/C$ and $D$, we use chromatic symmetric functions to provide a streamlined, combinatorial formula for the restriction to a specific maximal type $A$ parabolic subgroup. As an application, we prove a conjecture of Lesnevich regarding the representations associated with ideals of types $B$ and $C$ identified under the standard poset isomorphism. Specifically, we show that the corresponding dot action representations are isomorphic if and only if the ideals determine the same reflection sets in their respective root systems. Furthermore, we show that a Catalan number counts the total number of such ideals.

We prove that for any set $A$ with positive multiplicative upper density, there exist $x$, $y$, $z$ and $w$ in $A$ such that $x^2 + y^2w^2 = z^2$, generalizing a result of Frantzikinakis, Klurman and Moreira. Our techniques apply more broadly to a wide class of systems of equations, allowing us to solve these equations simultaneously and furthermore to restrict the w variable in an entirely different multiplicatively dense set. To do this, we prove a novel multiple recurrence result for multiplicative dynamical systems.

We prove an asymmetric version of the discretised ring theorem.

Let X and Y be finite ordered sets and let $X^Y$ denote the ordered set of order-preserving maps from $Y$ to $X$. Let $t$ be a~term formed from a~single variable by exponentiation. We prove that, for every term $t$ with at most twelve variable occurrences and arbitrary finite ordered sets $A,B$, $t(A)\cong t(B)$ implies that $A\cong B$. General constructions and reconstruction principles are developed first; the second part gives the occurrence-by-occurrence proofs. At twelve occurrences the proof covers 4,766 interchange normal forms, representing all 58,786 binary parenthesizations, including 40 exceptional forms. Two additional unbounded reconstruction families, complete finite audit data, and 457 exact height-comparison certificates are included.

The $k$-th power of a simple graph $G$, denoted $G^k$, is the graph with vertex set $V(G)$ where two vertices are adjacent if they are within distance $k$ in $G$. We investigate the zero forcing number of graph powers. Powers of graphs are much denser and generally not encompassed by existing results on zero forcing of graphs, hence their study requires a different approach. In contrast with the usual zero forcing behavior under edge deletion, we show that the zero forcing parameter (and variations of it) is monotone with respect to taking powers. We also determine the zero forcing number of powers of paths and cycles, together with upper bounds for powers of spiders and of grids. We then present spectral lower bounds on the zero forcing number of graph powers which uniquely use the spectrum of the base graph, as well as linear programming methods to compute these bounds. Finally, we study the sharpness of the derived bounds. To derive our results we use techniques ranging from graph theory, linear algebra and polynomial optimization.

Let $\cP_{\ell,k}$ be the set of rooted binary phylogenetic networks with $\ell$ labelled leaves, $k$ reticulations and no parallel edges. We write $|\cP_{\ell,k}|=W_k(\ell)+D_k(\ell)$, where the weighted count $W_k(\ell)$ adds the inverse orders of the leaf-fixing automorphism groups and the defect $D_k(\ell)$ collects the remainder. The weighted count satisfies, for every $k$, a recursion over the source layers of the tree-component structure that involves neither a list of component graphs nor any distinction between symmetric and asymmetric configurations, and its exponential generating function is a Laurent polynomial in $\sqrt{1-2x}$. Automorphism groups of networks are $2$-groups, elementary abelian for $k\le5$ but not in general. For $k\le5$ the defect is the weighted count of networks with a distinguished involution, which obeys an extension of the same recursion. An exact symbolic evaluation of the two recursions yields $|\cP_{\ell,k}|$ in closed form for $k\le5$, the case $k=5$ being new; it reproduces the published counts for $k\le4$ and corrects a coefficient in a published generating function for $k=3$. For every $k$, uniformly over explicit ranges of $k$, we prove that the non-tree-child networks are a fraction $2k(k-1)/\ell$ of the tree-child networks to leading order, which gives the third term of the asymptotic expansion of $|\cP_{\ell,k}|$. We also prove that reticulation-visible networks exceed tree-child networks by the fraction $k(k-1)/\ell$, and that a uniformly random network in $\cP_{\ell,k}$ has a nontrivial automorphism with probability $k(k-1)/(4\ell^3)$ to leading order.

We prove that every recovering pair $(\mathcal{A},\mathcal{B})$ of nonempty families of subsets of an $n$-element set satisfies $$|\mathcal{A}||\mathcal{B}|\le\left(\frac{223949}{100000}\right)^n=(2.23949)^n.$$ A result of Mond, Souza and Versteegen, combined with the sharp $(9/4)^n$ bound of Fang and Huang for cancellative pairs, gives $(2.2499)^n$ for recovering pairs. We improve this to $(2.23949)^n$. The proof combines two entropy inequalities. One is derived directly from the recovering property, while the other applies the Fang-Huang bound to families obtained from the original pair by a construction of Mond, Souza and Versteegen. The two inequalities are then combined by a convex combination.

Answering a question of Madill and Rampersad, we give an example a of a $2$-automatic sequence whose abelian complexity is not $2$-regular.

Let $\Fq$ be a finite field, let $F,G\in\Fq[T]$ be nonconstant, and let $J\in\Fq[T]$. We study the parameterized two-parameter family \[ C_{a,b}=\{(x,t)\in\Fq^2:t=J(x)+aF(x)+bG(x)+\lambda ab\}, \qquad (a,b)\in\Fq^2, \] where $\lambda\in\Fq^\times$, with incidences counted with parameter multiplicity. If $d=\deg F$, $e=\deg G$, and $\charac(\Fq)>d+e$, we prove the uniform spectral estimate \[ \left|I(\cP,\Omega)-\frac{|\cP||\Omega|}{q}\right| \ll_{d,e}q^{5/8}\sqrt{|\cP||\Omega|} \] for arbitrary $\cP,\Omega\subseteq\Fq^2$. The proof combines two additive Fourier transforms with a fourth-moment estimate for Kloosterman matrices. If $1\le d,e\le2$ and $\charac(\Fq)>3$, midpoint–difference coordinates and finite Weyl quantization sharpen the error to the Vinh scale $O(q^{1/2}\sqrt{|\cP||\Omega|})$. We further show that the latter phenomenon is intrinsic to the coefficient curve rather than to the degree of a chosen parametrization. There is a polynomial Lüroth factorization \[ F=f\circ H,\qquad G=g\circ H,\qquad \Fq(f,g)=\Fq(T), \] unique up to an affine change of $H$. If $\Gamma$ is the plane curve parametrized properly by $(f,g)$ and \[ \kappa_H=\max_{s\in\Fq}|H^{-1}(s)|, \] then finite-fibre pullback multiplies the centered incidence norm by at most $\sqrt{\kappa_H}$. Consequently, if $\deg\Gamma\le2$ and $\charac(\Fq)>3$, then \[ \left|I(\cP,\Omega)-\frac{|\cP||\Omega|}{q}\right| \ll \sqrt{\kappa_H}\,q^{1/2}\sqrt{|\cP||\Omega|}, \] even when $F$ and $G$ have arbitrarily large raw degree. The resulting expansion estimates for $J(x)+yF(x)+zG(x)+\lambda yz$ hold without lower bounds on the individual input sets. In the balanced regime, the $q^3$ missing-value scale for this family is already covered by a theorem of Arala–Chow; the incidence theorems here additionally yield expansion consequences in strongly unbalanced regimes.

For a graph $H$, its $k$-colour vertex Ramsey class is the set of all graphs $G$ such that any colouring of the vertices of $G$ in $k$ colours results in a monochromatic (induced) copy of $H$. We prove that for any $k$, Ramsey classes of any non-isomorphic graphs are distinct.

The theory of cluster algebras is closely connected to the theory of total positivity; indeed, the desire to better understand total positivity was one of the main motivations for Fomin and Zelevinsky's introduction of cluster algebras [arXiv:math/0104151]. In particular, any cluster variety whose coordinate ring has a cluster structure has a natural notion of positive part: the subset of the variety where all cluster variables are positive. In this paper, we explain that there are other signed cells contained in cluster varieties that are equally natural from a cluster-theoretic point of view. These come from signed seeds, which can be thought of as a $\mathbb{Z}/2\mathbb{Z}$-grading on cluster variables, and which were introduced in [arXiv:2310.17727] in the context of the amplituhedron. More generally, given any abelian group $G$, we introduce the notion of a $G$-graded seed for a cluster algebra, which is a way of assigning elements of $G$ to each cluster variable which is compatible with the cluster structure. When $G$ is the multiplicative group $\{-1, 1\}$, this recovers the above notion of signed seed; when $G = \mathbb{C}^*$, this recovers the notion of cluster automorphism group [GSV10] or cluster dilation group [arXiv:2603.17890]; and when $G = \mathbb{Z}^d$, this recovers the notion of graded cluster algebra studied by Grabowski-Launois [arXiv:1301.2133], Grabowski [arXiv:1309.6170] and Gekhtman-Shapiro-Vainshtein [GSV10, Section 5.2] (which had previously appeared in special cases in work of Fomin-Zelevinsky [arXiv:math/0602259]). The examples we study include the space of square matrices, symmetric matrices, skew-symmetric matrices, positroid varieties, and amplituhedron tiles. We also connect this notion to tropical mutation when $G = \mathbb{R}$ or $\mathbb{Z}$.

Our combinatorial analysis is motivated by the PDE dynamics \begin{equation} \mathbf{u_t} = \mathbf{u_{xx}} + \mathbf{g}(\mathbf{u}),\qquad 0<\mathbf{x}<1, \end{equation} of solutions $\mathbf{u}=\mathbf{u}(\mathbf{t},\mathbf{x}),\ \mathbf{t}\geq 0$, under Neumann boundary conditions. For dissipative nondegenerate nonlinearities $\mathbf{g}$, the global attractors $\mathcal{A}=\mathcal{A}_\mathbf{g}$ of the PDE can then be classified by the orderings of their $2n+1$ equilibria $\mathbf{v}$ at the boundaries $\mathbf{x}=0,1$. We encode the boundary orders as Hamiltonian Sturm permutations. The name ”Sturm” refers to nodal properties of PDE solutions $\mathbf{u}(\mathbf{t},\mathbf{x})$. ”Hamiltonian” refers to the second order pendulum ODE for equilibria $\mathbf{v}(\mathbf{x})$: \begin{equation} 0 = \mathbf{v_{xx}} + \mathbf{g}(\mathbf{v}). \end{equation} We determine the generating function $a(z)=\sum_n a_nz^n$ for the counts $a_n$ of Hamiltonian Sturm permutations. For $n\rightarrow\infty$, this provides explicit asymptotics of $a_n$. We refine these counts as $a_n=\sum b_{rq}$. Here $b_{rq}$ counts Hamiltonian Sturm permutations with $2r+1$ spatially homogeneous equilibria and $2q$ spatially non-homogeneous equilibria, such that $r+q=n$. We also determine the explicit generating function $b(x,y)=\sum_{r,q} b_{rq}x^ry^q$. This implies asymptotically Gaussian distributions of the probabilities $p_{nr}=b_{rq}/a_n$ with $r+q=n$, asymptotically for large $n$. We derive asymptotics for means and variances, with error estimates of order $1/n$. All asymptotics are based on work by Flajolet and Sedgewick. We conclude with numerical illustrations and remarks on nonlinearities $\mathbf{g}(\mathbf{u},\mathbf{u_x})$ under periodic boundary conditions $\mathbf{x}\in\mathbb{S}^1=\mathbb{R}/2\mathbb{Z}$, where rotating waves arise.

We study the average sizes of ideals and antichains in several families of posets arising from rooted unlabeled trees. For ideals, we encode order ideals by coloring vertices red, with the condition that every descendant of a red vertex is also red. Using generating functions and singularity analysis, we show that for every family governed by a quadratic root decomposition, the average number of red vertices in a tree of size $n$ is asymptotic to $n/3$. This includes binary plane trees, plane 1-2 trees, plane 2-trees, and 0-1-trees, while unrestricted rooted plane trees instead have average ideal size asymptotic to $2n/5$. For antichains, represented by pairwise incomparable blue vertices, the corresponding asymptotic constants depend on the particular branching function; for example, binary plane trees have average antichain size asymptotic to $n/6$. We also consider plane trees in which every vertex has at most $k$ children and prove that the asymptotic proportion of red vertices increases strictly with $k$, from $1/3$ when $k=2$ to the limiting value $2/5$ for unrestricted rooted plane trees. The asymptotic proportion of blue vertices also increases strictly with $k$, tending to $3/10$. The proofs are based on bivariate generating functions together with the analytic implicit-function and smooth implicit-function methods of analytic combinatorics.

Why are some dynamic lot-sizing problems polynomial? We address this question by introducing Grid Theory, a structural framework based on cumulative production and the additive structure of production bounds. For a general single-item dynamic lot-sizing model with lower and upper production bounds, there exists an optimal extreme solution in which, within each regeneration interval, all but at most one production quantity lie on a boundary value. This induces additive grids, and the Main Grid Theorem establishes that an optimal cumulative production trajectory can be restricted to these discrete sets. Although the resulting grids may be exponentially large, we introduce the notion of additive dimension to capture production-bound profiles whose boundary sums admit a low-dimensional representation. We show that bounded additive dimension yields a polynomially constructible grid envelope and a polynomial time grid-based dynamic programming algorithm. The framework extends to separable concave costs and establishes polynomial solvability of several families, including constant capacities, minimum order quantities, a fixed number of capacity levels, fixed-degree polynomial capacities, periodic capacities, and piecewise polynomial capacities. In particular, polynomiality may hold even when the number of distinct capacity values grows with the planning horizon. Grid Theory thus identifies additive structure, rather than the number of distinct resource values, as a sufficient mechanism for polynomial solvability.