Papers

We give a structural account of the finite free convolutions of Marcus, Spielman, and Srivastava in terms of reproducing kernel inner products on polynomial spaces and a multilinear model over the squarefree algebra. In this model, additive convolution becomes algebra multiplication, and the nilpotent logarithm linearizes it, recovering the finite free cumulants of Arizmendi and Perales. This perspective leads to a class $\mathcal{LC}_n$ of multilinear polynomials characterized by nonpositivity of higher-order cumulants, closed under additive convolution and satisfying several key permanence properties associated with negatively dependent measures. We show that every graph Laplacian pencil belongs to this class, with higher-order cumulants given by Hamiltonian cycle counts in induced subgraphs.

We study Euler characteristics of tropical subvarieties of tropical abelian varieties. We prove that every H-regular subvariety, locally modeled on tropicalizations of sufficiently well-behaved very affine varieties, has nonnegative signed Euler characteristic. This gives a tropical analogue of a theorem of Green-Lazarsfeld for subvarieties of complex abelian varieties. The main input is a local vanishing theorem for H-regular tropical fans, which also yields a Lefschetz-type theorem for affine H-regular tropical varieties. We further show that the signed Euler characteristic inequality fails for general tropical subvarieties of tropical abelian varieties, and we construct a 3-dimensional tropical fan whose link is not homotopy equivalent to a bouquet of 2-spheres.

The Dyn-Farkhi conjecture asserts that the square of the Hausdorff distance from a compact set to its convex hull is subadditive with respect to Minkowski addition. The conjecture is elementary in dimension 1, was recently proved by Meyer in dimension 2, and was disproved in dimensions $n\geq3$ by Fradelizi, Madiman, Marsiglietti, and Zvavitch. The symmetric case $A=B$, however, remained open. We show that the conjecture already fails in this restricted setting. More precisely, for every $n\geq3$, we construct a compact set $A\subset\mathbb{R}^n$ such that $$d(A(k))=d(A)>0$$ for every $1\leq k\leq n-1$, where $d(X)$ is the Hausdorff distance from $X$ to its convex hull and $A(k):=\frac1k (A+\dots+A)$ is the $k$-fold iterated Minkowski average of $A$. We also prove that the threshold $k=n$ is sharp: for every nonempty compact $A\subset\mathbb{R}^n$ with $n\geq 2$, we have $$d(A(n))\leq \left(1-\frac{n-1}{n(2n-1)}\right)d(A).$$

Let $\Gamma = (G, \sigma)$ be a signed graph and $v$ a non-isolated vertex of $\Gamma$. Let $\Gamma-v$ denote the graph obtained by deleting the vertex $v$ together with all signed edges incident to it from $\Gamma$, and $d_{\Gamma}(v)$ the degree of $v$ in $\Gamma$. In this paper, we prove that the largest eigenvalue $\lambda_1(\Gamma)$ of $\Gamma$ satisfies \[ \lambda_1(\Gamma) \le \sqrt{\lambda_1^2(\Gamma - v) + 2d_\Gamma(v) - 1}, \] and we also present a refined version of this bound. Moreover, we characterize the extremal signed graphs achieving equality when $\Gamma$ is connected and $d_\Gamma(v)\ge 2$, which are switching equivalent to the balanced complete signed graph.

There are up to isomorphism exactly three strongly regular graphs with parameters $(120,56,28,24)$ whose automorphism group acts primitively on the vertices. Two of these graphs belong to classical families: one is the non-orthogonality graph on anisotropic points of the hyperbolic quadric $\mathcal Q^+(7,2)$, and the other one belongs to the Johnson scheme. The third one is not well understood. In this paper, we give a description of this graph in terms of ovoids and spreads of $\mathcal Q^+(7,2)$, or equivalently in terms of overlarge sets of Steiner systems with parameters $(3,4,8)$.

We derive a generating function identity for the joint distribution of the numbers of peaks of a permutation and its inverse, via enriched $P$-partitions. The coefficients of the corresponding peak polynomial $W_n(s,t)$ satisfy a second-order recurrence. A martingale formulation of this recurrence yields a bivariate central limit theorem, showing that the two statistics are asymptotically independent. We also give an exact closed form for their covariance, which is of order $n^{-1}$.

In 2023, Greaves, Syatriadi, and Yatsyna found a set of $57$ equiangular lines in $\mathbb{R}^{18}$, breaking the previous record. In 2025, Lin, Munemasa, Taniguchi, and Yoshino constructed a large number of sets of $57$ equiangular lines in $\mathbb{R}^{18}$ as affine equiangular sets in an integral overlattice of $A_9^2 \oplus A_1$. In this paper, we construct further sets of $57$ equiangular lines in $\mathbb{R}^{18}$ from Latin squares of order $6$ and Pasch configurations, realized as affine equiangular sets in an integral overlattice of $A_5^3 \oplus A_1^4$. Unlike the previously known examples, these sets are not strongly maximal. Moreover, some of them have only five distinct Seidel eigenvalues, fewer than any previously known examples.

Godsil and Sun asked whether, for a strongly regular graph $X$ and any two different edges $e$ and $f$, the edge-deleted graphs $X\setminus e$ and $X\setminus f$ are degree-similar. We give an affirmative answer to the problem of Godsil and Sun. In fact, we prove the stronger statement that if $X$ is a $1$-walk-regular graph, then for any two edges $e$ and $f$ of $X$, the graphs $X\setminus e$ and $X\setminus f$ are orthogonally degree-similar. The proof is based on an edge version of the orthogonal-intertwiner method: the equality of the Gram matrices of the projected endpoint vectors in every eigenspace yields an orthogonal matrix commuting with the adjacency matrix and sending one pair of ordered endpoint vectors to the other.

We prove a conjecture of D. B. Shapiro, which asserts that no integral [10,10,16] formula can arise as a restriction of any Hurwitz-Radon formula.

In 2025 Cornet, Dravec, and Torres determined the domination number $\gamma(J(n, 3))$ of the Johnson graph for every even $n \ge 6$, expressing it as a closed form $\phi_n$ in terms of Fort\textendash{}Hedlund covering numbers, and conjectured the same value for odd $n$. We prove this conjecture: $\gamma(J(n, 3)) = \phi_n$ for every odd $n \ge 7$, completing the determination of $\gamma(J(n, 3))$ for all $n \ge 6$.

We present several Laplacian-type matrices associated with a loopless digraph $D$: the out-/in-degree Laplacians $\mathcal L_{\mathrm{out}},\mathcal L_{\mathrm{in}}$, the incidence Laplacian $\mathcal L_{\mathrm{inc}}=BB^{\mathsf T}$, and the symmetrized and skew-symmetrized variants $\mathcal S_{\mathrm{out}},\mathcal K_{\mathrm{out}}$. We show that $\mathcal L_{\mathrm{inc}}(D)$ coincides with the Laplacian of the underlying undirected multigraph, and we derive spectral and characteristic-polynomial relations under arc reversal and complementation (including a simplification for Eulerian digraphs for $\mathcal S_{\mathrm{out}}$). We demonstrate that the spectral radius of $\mathcal L_{\mathrm{out}}$ is bounded above by the order of the digraph and give a characterization in the equality case. We further obtain explicit formulas for joins and line digraphs, giving a general determinantal identity relating the out-degree Laplacian characteristic polynomials of a regular digraph and its line digraph.

Fix a prime $p$ and a constant $\frac{1}{p}<\alpha\leq 1$. Consider the random directed Erdős–Rényi bipartite graph $\vec G(n,\lceil\alpha n\rceil ,v)$ with bipartition $(V_1,V_2)$ of sizes $|V_1|=n$ and $|V_2|=\lceil\alpha n\rceil$, and edge probability $0<v<1$. Bhargava, DePascale and Koenig conjectured a limiting distribution for the $p$-Sylow subgroup of the sandpile group of $\vec G(n,\lceil\alpha n\rceil,v)$ as $n\to\infty$. We prove this conjecture. Similar results have previously been proved by computing the expected number of surjections from the random abelian $p$-group onto $H$, for each finite abelian $p$-group $H$. However, in the case of $p$-Sylow subgroups of sandpile groups of random directed bipartite graphs, these surjective moments often diverge to infinity, despite the conjectured limiting distribution having finite moments. We resolve this by restricting to a high-probability subset of graphs on which the surjective moments are well-behaved, and discarding a rare exceptional set of graphs whose contribution to the distribution vanishes but whose contribution to the surjective moments often diverges. Computing the conditional surjective moments on the good set and applying Wood's universality theorem yields the desired convergence in distribution.

We show that the partitions that always appear with nonnegative $e$-coefficients in chromatic symmetric functions of finite graphs are precisely the hook partitions.

In this article, we prove irreducibility results for a family of Markov chains arising in the study of redistricting and detecting gerrymandering. These chains use ReCom moves as their transition mechanism and are commonly employed in Markov chain Monte Carlo methods to generate ensembles of districting plans. Such ensembles are frequently used for outlier analysis, in which a proposed districting map is compared against the ensemble to determine whether it behaves atypically; this methodology often appears in expert testimony in redistricting litigation. We show that when the underlying dual graph is a triangular subset of the triangular lattice with side length 5 or larger, and each district consists of two merged geographic regions, the associated ReCom chain is irreducible. This provides another entry in the very small list of known classes of ReCom chains for which irreducibility has been established. We also demonstrate the fragility of this phenomenon by constructing an infinite family of maps for which the corresponding ReCom chain is not irreducible. Indeed, we produce a districting map that, after implementing a single ReCom move, always yields the same original map. These examples remain structurally close to the triangular lattice: they arise as subdivisions of the triangular lattice, and the resulting graphs have maximum degree at most 8. Finally, we prove irreducibility for a further special case: the ReCom chain on a 3 x n grid graph partitioned into three districts of size n.

The number of perfect matchings in a user-behavior bipartite graph is a natural measure of anonymity: more matchings mean greater uncertainty for an attacker. A fundamental question is which graph structure maximizes this count for a fixed infrastructure cost, represented by the number of edges. We prove that the answer is $k$-anonymity by partitions. Using Brègman's Theorem, we show that partitioning users into equal-sized groups and making each group a complete bipartite component achieves the theoretical upper bound on perfect matchings. For edge counts where an exact partition is impossible, we construct a family of graphs that asymptotically attains this bound as the group size grows. We further prove that this optimality is robust: after an attacker de-anonymizes a user by the most damaging choice, the resulting graph is still a partition graph and remains optimal. Together, these results provide a combinatorial justification for the widespread use of $k$-anonymity by partitions in anonymity system design.

Dyck paths and certain black/white labelling of nodes leads to the white-height. Using generating functions and tricks of the trade, we establish that the average white-height among Dyck paths of half length $n$ is asymptotic to $\frac12\sqrt{\pi n}$ for two different models. These are appealing results that could be presented to students to learn the trade. A last section links walks with double steps and 2-Motzkin paths. For them, the white-height is the natural concept. Folks who might be offended by the notion of white-height might choose their own colours that they like.

For graphs $F$ and $H$, the generalized Turán number $\mathrm{ex}(n,F,H)$ denotes the maximum number of copies of $F$ in an $H$-free graph on $n$ vertices. We prove that if $s\in \{2,3\}$, $s< a\leq b$ and $t$ is sufficiently large, then $\mathrm{ex}(n,K_{a,b},K_{s,t})=\Theta(n^s)$. The $s=2$, $a=b=3$ case of this result answers a question of Spiro. Proving another conjecture of Spiro, we show that for every graph $F$ with at least one edge, there exist infinitely many real numbers $r$ such that $\mathrm{ex}(n,F,H)=\Theta(n^r)$ holds for some graph $H$.

Dedekind's problem counts monotone Boolean functions, equivalently downsets of a Boolean lattice. We recast this enumeration as a finite layer-ratio reconstruction problem for the Whitney numbers of the ranked ideal lattice. An exact adjacent-layer double count expresses each layer ratio through local averages of the number of addable elements and the number of removable elements. Reversible fixed-layer Markov chains estimate these averages and hence estimate the Dedekind number $M(n)$. Backtests at $M(8)$ and $M(9)$ calibrate seed-level variability under the fixed protocol and measure the observed Monte Carlo budget scaling. The resulting estimate probes the Whitney-number sequence of the ideal lattice. Although these rows have previously been described empirically as unimodal, the high-precision $n=9$ estimate has a shallow two-shoulder feature around the central rank, contrary to that empirical description; $n=11$ and $n=13$ center-window estimates show a larger-contrast analogous pattern. The protocol estimate for $M(10)$ is \[ \widehat M(10)=(8.9360\pm0.0010)\times 10^{78}, \] where the displayed uncertainty is the budget-based forecast scale from the cross-$n$ scaling law under the production budget.

A biclique decomposition of a graph is a partition of its edges into complete bipartite subgraphs. We consider graphs whose vertices can be ordered such that the neighborhood of every vertex is the union of a sublinear number of intervals. We observe that these graphs admit compact representations in the form of biclique decompositions of small size. Here, the size of a decomposition is measured as the sum of the number of vertices of its bicliques. Combining this result with the existence of suitable vertex orderings for graphs of low neighborhood complexity, as proven by Welzl in 1988, we recover and extend several known results up to logarithmic factors. These results include upper bounds on the Zarankiewicz problem, matrix multiplication, quantum circuit complexity, and shortest path algorithms in "well-structured" instances.

We study $\mathbb{Z}$-Ramsey ultrafilters, ultrafilters containing witnesses to every shift-invariant instance of Ramsey's theorem. We prove that it is consistent that there are no $\mathbb Z$-Ramsey ultrafilters. We also prove that every $(\mathbb{Z},3)$-Ramsey ultrafilter, as well as every $\mathbb{Z}$-Ramsey P-point, is selective. Further, we exhibit a generic extension – using quotient algebras of the form \(\mathcal{P}(\mathbb{Z})/\mathcal{I}\) for certain \(F_{\sigma }\)-ideals – that contains P-points that are not \(\mathbb{Z}\)-Ramsey ultrafilters, thereby addressing open questions raised by Petrenko and Protasov.

Lovász' famous Hamiltonicity conjecture (1969) states that every connected vertex-transitive graph has a Hamiltonian path. A stronger version of the conjecture, often attributed to Thomassen (1978), states that every sufficiently large such graph even has a Hamiltonian cycle. Despite the great amount of attention these conjectures have attracted over the past decades both in the combinatorial and algebraic communities, for more than 40 years the best known lower bound for the maximum length of a cycle (path) in a connected vertex-transitive graph of order $n$ remained of the form $\Omega(\sqrt{n})$, due to Babai (1979). A series of recent works has successively improved the exponent in this lower bound further. In this paper, improving the previous state-of-the-art bound $\Omega(n^{9/14})$ due to Norin et al.~(2025), we prove that every connected vertex-transitive graph of order $n$ contains a cycle of length at least $n^{2/3-o(1)}$. This hits a natural barrier for several existing approaches from previous work. Our proofs combine recent embedding techniques for paths in sublinear expanders, sublinear expander decompositions of almost-regular graphs, and several additional combinatorial ideas.

The triangle packing number $\nu(G)$ of a graph $G$ is the maximum size of a set of edge-disjoint triangles in $G$. Tuza conjectured that in any graph $G$ there exists a set of at most $2\nu(G)$ edges intersecting every triangle in $G$. We show that Tuza's conjecture holds in the random geometric graph for a large range of densities. We also study the problem of covering almost all edges of the random geometric graph with edge-disjoint copies of some fixed graph $F$. In particular, we show the existence of almost-perfect packings for an infinite family of $F$, and state some negative results as well.

For a clique $K_r$, a graph is $K_r$-saturated if it contains no copy of $K_r$ and the addition of any edge from its complement creates a $K_r$. A classical result of Erdős-Hajnal-Moon and Zykov shows that the number of edges of an $n$-vertex $K_r$-saturated graph is at least $(r-2)n-\binom{r-1}{2}$. In this paper, we focus on the number of edges of the $K_r$-saturated graphs with a fixed matching number. Let $G$ be an $n$-vertex $K_r$-saturated graph with matching number $\nu(G) = s$. For sufficiently large $n$, we prove that the number of edges \begin{equation*} e(G)\geq \left\{\begin{array}{cl}{(r-1)n-\frac{r}{2}(r-1)-1,}&{\quad\mathrm{if}~s=r-1;}\\{(r-1)n + (s-r)^2 - \frac{1}{2}(r+2)(r-3) - 5,}&{\quad\mathrm{if}~s>r-1.}\\\end{array}\right. \end{equation*} Moreover, we completely characterize the graphs attaining the equality.

We present Trellis: an autoformalization system that leverages LLM agents in a deterministically constrained workflow to enforce incremental progress in Lean autoformalization tasks through iterative refinement of natural language proofs. Our approach is motivated by the common mathematician's notion of what it means to have a rigorous proof in the first place: namely, that it would be routine to elaborate any part of the proof in further detail. The result is a system which aims to achieve reliable autoformalization on a modest budget and with generalist agents, with specialization to autoformalization coming not from any task-specific agent training but instead from a meaning-of-rigor inspired workflow enforced by process semantics. We link to an end-to-end Lean formalization of a recent Ramsey theory breakthrough produced by the process.

We consider the $SO(d)$ and $O(d)$ invariant sectors of the bosonic $d$-matrix harmonic oscillator with $U(N)$ gauge symmetry. The micro-canonical degeneracy $\mathcal{Z}( N , d , k )$ for fixed energy $k$ is expressed as a pairing between an $N$-dependent vector and a $d$-dependent vector in the space of partitions of the integer $k$. This pairing formula is derived by counting invariant words in multi-matrix variables $X^i_{j,a}$, using properties of Clebsch-Gordan multiplicities (Kronecker coefficients) for the symmetric group $S_k$, Schur-Weyl duality and harmonic analysis on the homogeneous space $U(d)/SO(d)$. Analytic formulae for large $N$ and $k$ with $ k \le N $ are obtained using group integrals over $U(N)$ and $SO(d)$ (or $ O(d)$). The micro-canonical heat capacity in this regime is negative and turns positive, at a critical value $k_{\rm crit}$, due to finite $N$ modifications to the counting, thus forming what we denote as a characteristic caloric fold in the $ E $ versus $T$ curve. Data from the pairing formula is well fitted by $k_{\rm crit} \sim { N^2 \over 4 }$ for small values of $d$. A derivation of this large $N$ formula is given using a matrix model approximation and semi-classical analysis of the eigenvalue density. The large $N,d$ limit of the degeneracies reveals a key role for ribbon graph combinatorics. The caloric fold is also notably a property of black hole thermodynamics in anti-de-Sitter spaces. We propose the spherically symmetric \(SO(d)\) and \(O(d)\) invariant sectors of \(d\)-matrix quantum mechanics as tractable matrix systems for capturing key features of dual descriptions of black-hole thermodynamics.

For a graph $F$, let $h_F(n,q)$ be the minimum number of copies of $F$ in an $n$-vertex graph with $\mathrm{ex}(n,F)+q$ edges, where $\mathrm{ex}(n,F)$ is the maximum number of edges in an $n$-vertex $F$-free graph. Let $c(n,F)$ be the minimum number of copies obtained by adding one edge to an extremal $F$-free graph. Mubayi's supersaturation conjecture predicts, under a stability hypothesis, that $h_F(n,q)\ge q\,c(n,F)$. Ma and Yuan recently constructed stable graph counterexamples for every fixed $q\ge4$; they asked whether the one-edge equality $h_F(n,1)=c(n,F)$ might still hold for every graph $F$ containing a cycle. We give a negative answer to their question. For each integer $t\ge6$, let $H_t$ be obtained from the $t$-vertex path by replacing each edge with a $3t$-page book, using disjoint page vertices for different path edges. Then $h_{H_t}(n,1)<c(n,H_t)$ for infinitely many values of $n$. Moreover, by taking $t$ large, the ratio $h_{H_t}(n,1)/c(n,H_t)$ can be made arbitrarily small along infinitely many values of $n$.

Let $f(n,H)$ be the maximum number of copies of a graph $H$ in a planar graph of order $n$. When $H$ is a connected graph on four vertices, $f(n,H)$ has been completely determined except for two cases: $K_{1,3}^+$ (the claw graph $K_{1,3}$ with one additional edge) and $K_4^-$ (the complete graph $K_4$ with one edge removed). Here, we address these two cases and establish that for all $n\ge4$, $$f(n,K_{1,3}^+) = 4n^2-12n-4 \textrm{ and } f(n,K_4^-) =\frac{1}{2}(n^2+9n-40).$$

Resolving in a strong sense a problem of Gyárfás on the union of two perfect graphs, we prove that for every pair of positive integers $d$ and $k$, there is a graph $G$ with clique number $k$ and chromatic number $k^d$ that is the union of $d$ comparability graphs. We also show that the chromatic number can be replaced by the fractional chromatic number or $\frac{|V(G)|}{\alpha(G)}$.

Ballantine, Beck, Merca, and Sagan conjectured four identities, collectively Conjecture 19, relating the image of the map pre_k on integer partitions to four OEIS sequences. We prove parts (i) and (iii) unconditionally, prove part (iv) unconditionally using the injectivity of pre_2 on partitions of n (Conjecture 1 of the same paper, proved by Li in arXiv:2508.00971), and show that this injectivity is in fact equivalent to part (iv). For part (ii) we prove the partition-theoretic half unconditionally and reduce the remaining content to a 2006 conjecture of Dean Hickerson on the OEIS concerning Huffman coding. We also correct a sign error in the published statement of part (iii): the correct identity is chi(ImP_3(n)) = A213213(n) - 1, not 1 + A213213(n) as stated.

We study partial orders induced by constrained variants of finite graph homomorphisms: monomorphisms, embeddings, full homomorphisms, vertex-surjective, edge-surjective and surjective homomorphisms, and locally injective, locally surjective and locally bijective homomorphisms. For each order we ask for analogues of the standard structural properties of the graph homomorphism order: canonical cores, past- or future-finiteness, universality, gaps and finite dualities. The comparison shows which phenomena are specific to ordinary homomorphisms and which are consequences of simpler order-theoretic mechanisms. We identify cores for full and surjective homomorphisms, relate full-homomorphism cores to point-determining graphs, characterize gaps in the full homomorphism order, and give finite obstruction bounds for several one-sided finite orders. We also analyze locally constrained homomorphisms on connected graphs. In particular, locally injective homomorphisms have all connected graphs as cores, admit infinite-chain density under natural degree-refinement assumptions, have explicit gap witnesses, and are universal already on finite connected bipartite subcubic cactus graphs. The paper reorganizes and extends several earlier arguments into a single framework for constrained homomorphism orders.