Papers

We study the problem of efficiently finding large common induced subgraphs of two independent Erdős–Rényi random graphs $G_1, G_2 \sim \mathbb{G}(n,1/2)$. Recently, Chatterjee and Diaconis showed that the largest common induced subgraph of $G_1$ and $G_2$ has size $(4-o(1))\log_2 n$ with high probability. We first show that a simple greedy online algorithm finds a common induced subgraph of $G_1$ and $G_2$ of size $(2-o(1)) \log_2 n$ with high probability. Our main result shows that no online algorithm can find a common induced subgraph of $G_1$ and $G_2$ of size at least $(2+\varepsilon) \log_2 n$ with probability bounded away from $0$ as $n \to \infty$. Together, these results provide evidence that this problem exhibits a computation-to-optimization gap. To prove the impossibility result, we show that the solution space of the problem exhibits a version of the (multi) overlap gap property (OGP), and utilize an interpolation argument recently developed by Gamarnik, Kizildağ, and Warnke that connects OGP and online algorithms.

We give an expression for the frequency of non-simple curves in closed surfaces and exploit it to study relative frequencies of such curves in large genus. This extend to the case of non-simple curves Mirzakhani's expressions of frequencies in terms of Konsevitch polynomials and Delecroix-Goujard-Zograf-Zorich large genus asymptotics for those frequencies. In particular, with K fixed, we identify which types of curves with K intersections are most common.

Erdős and Hajnal proposed a problem that: is it true that every $(2n+1)$-vertex graph with $n^2+n+1$ edges contains two vertices of equal degree connected by a path of length three? The edge bound is sharp by the complete bipartite graph $K_{n,n+1}$. Recently, Chen and Ma [Journal of Combinatorial Theory, Series B, 179:1-18, 2026] answered this problem affirmatively for every $n \ge 600$. In the same paper, they further conjectured that for sufficiently large $n$, the statement is true if we replace the path of length three by a path of fixed odd length. In this paper, we confirm their conjecture.

Let $H$ be obtained from a cyclically $4$-edge-connected cubic planar graph $Y$ other than $K_4$ by deleting two adjacent vertices. We provide a short proof that if $H$ has circumference at least $k$ for some even integer $k \ge 4$, then $H$ contains a cycle of length between $k$ and $3k/2$. As a consequence, we show that the line graph $G$ of $Y$ contains a cycle of length $l$ avoiding any prescribed vertex of $G$, for every $l \in \{3\} \cup \{5, \dots, |V(G)| - 1\}$. The proofs integrate Euler's formula and the Three Edge Lemma, established by Thomas and Yu, and independently by Sanders, in a novel way. This work was partially motivated by conjectures of Bondy and Malkevitch.

We consider unimodular matrices $M$ such that neither $M$ nor $M^{-1}$ contain zero entries. Matrices typically exhibit a trade-off: small $M$ imply large $M^{-1}$. We investigate rare cases where both remain small, classify these matrices up to symmetry, and discuss aspects of this balanced setting.

Let $\Sym{n}$ denote the set of all permutations on $n$ labels. Let $c:[0, 1]^2\to [0, \infty)$ be a twice continuously differentiable function. A subfamily of the Mallows model is the Gibbs probability measures on $\Sym{n}$ such that $\mathbb{P}(X=\sigma)=L_n^{-1} \prod_{i=1}^{n}\exp(-c(i/n, \sigma(i)/n))$. Mukherjee [Ann. Stat., Vol. 44(2), pp 853–875 (2016)] computed the limit of the log partition function and showed that $\lim_{n\to \infty}\frac{1}{n}\log L_n=-\Gamma_0$ where $\Gamma_0$ is the optimal cost associated with an entropy regularized optimal transport problem. In the KRP Memorial Volume of the Indian Journal of Pure and Applied Math, Pal conjectured an exact value for the limit $\lim_{n\to \infty} e^{-n\Gamma_0}L_n$ in terms of the Fredholm determinant of an integral operator and provided a partial proof. We give a complete proof of Pal's conjecture.

Threshold graphs are graphs that can be characterized in a number of different ways. For example, they are graphs that are $P_4,\ C_4,\ 2K_2$–free. They may also be characterized by a finite sequence of positive integers $a_1, \ldots, a_r$, such that $a_1\geqslant 2$ and $a_1 + a_2 + \cdots + a_r = |V(G)|$. Threshold graphs have the remarkable property that all graphs of the same order share a common integer Laplacian eigenbasis. This property characterizes threshold graphs. This result was proved in \cite{MachareteDelVecchio}. We give a different proof of the same result.

We study the realization map of deep ReLU networks, focusing on when a function determines its parameters up to scaling and permutation. To analyze hidden redundancies beyond these standard symmetries, we introduce a framework based on weighted polyhedral complexes. Our main result shows that for every architecture whose input and hidden layers have width at least two, there exists an open set of identifiable parameters. This implies that the functional dimension of every such architecture is exactly the number of parameters minus the number of hidden neurons. We further show that minimal functional representations can still have non-trivial parameter redundancies. Finally, we establish a generic depth hierarchy, whereby for an open set of parameters the realized function cannot be represented generically by any shallower network.

For a graph $G$ with $n$ vertices and a positive integer $k \leq n$, let $s_k(G)$ be the number of subtrees (subgraphs that are trees, not necessarily induced) of $G$ with $k$ vertices. The subtree polynomial of $G$ is $S(G;x) = \sum_{k=1}^n s_k(G) x^k$. In this paper, we consider dense connected graphs with a minimum degree that is linear in the number of vertices. We prove that the number of missing vertices in a random subtree is asymptotically Poisson-distributed and deduce that all the roots of the subtree polynomial have to be close to $0$.

A problem dating back to Boole [Laws of Thought, Walton & Maberly,1854] is what can be computed about the probability of a finite union of events when given as input the probabilities of intersections of some of the events. The modern geometric study of the problem can be traced back to Hailperin [Amer. Math. Monthly 2 (1965) 343–359] who phrased the problem in the language of linear programming and generalized it to logical formulas of the events other than disjunction, heralding a substantial body of work in probabilistic logic [Nilsson, Artif.\ Intell.\ 28 (1986) 71–87], including the probabilistic satisfiability problem of Georgakopoulos, Kavvadis, and Papadimitriou [J.Complexity 4 (1988) 1–11], as well as fundamental connections to the geometry of metrics via cut and correlation polytopes [Deza and Laurent, Geometry of Cuts and Metrics, Springer, 1997] and to the study of marginal polytopes in graphical models of machine learning [Wainwright and Jordan, Found.\ Trends Mach.\ Learn. 1 (2008) 1–305]. This paper (i) describes the pertinent geometry of Boole's problem via coordinate projections of an elementary polytope arising essentially from Hailperin's linear program on the atoms of a Venn diagram, and (ii) shows that computing the optimal interval for the union probability is NP-hard, resolving an apparent gap in the literature highlighted by Pitowsky [Math.\ Programming 50 (1991) 395–414] and Boros et al. [Math.\ Oper.\ Res. 39 (2014) 1311–1329 and 51 (2026) 134–148].

Let $A = \{a_1, \ldots, a_k\}$ be a nonempty finite subset of an additive abelian group $G$. For a nonnegative integer $h$, the \emph{$h$-fold signed sumset} of $A$, denoted by $h_{\pm} A$, is defined by $$ h_{\pm} A = \Biggl\{\sum_{i = 1}^{k} \lambda_i a_i : \lambda_i \in \{-h, \ldots, h\}, \ \sum_{i = 1}^{k} |\lambda_i| = h \Biggr\}, $$ and the \emph{restricted $h$-fold signed sumset}, denoted by $h_{\pm}^\wedge A$, is defined by $$ h_{\pm}^\wedge A = \Biggl\{\sum_{i = 1}^{k} \lambda_i a_i : \lambda_i \in \{-1, 0, 1\}, \ \sum_{i = 1}^{k} |\lambda_i| = h \Biggr\}. $$ We study direct and inverse problems for these signed sumsets, namely determining extremal bounds for their sizes and characterizing the structure of sets $A$ attaining these bounds. While such problems have been extensively studied and resolved in the additive group of integers, comparatively little is known in general abelian groups, especially for restricted signed sumsets. In this paper, we investigate the signed sumset $h_{\pm} A$ in arbitrary (not necessarily finite) abelian groups under the condition $A \cap (-A) \neq \varnothing$. We further analyze both $h_{\pm} A$ and $h_{\pm}^\wedge A$ when $A \cap (-A)$ has a prescribed size. These results are extended to generalized signed sumsets $H_{\pm} A = \bigcup_{h \in H} h_{\pm} A$, where $H$ is a finite set of nonnegative integers, with particular attention to $[0,h]_{\pm} A$. Furthermore, using the polynomial method, we establish nontrivial lower bounds for $|h_{\pm}^\wedge A|$ in arbitrary fields. In addition, for $h = 2, 3, 4$, we derive lower bounds for $|h_{\pm} A|$ in arbitrary fields under the condition $A \cap (-A) = \varnothing$.

The Helmholtzian matrix of a graph $G=(V(G),E(G))$ is a graph-theoretic analogue of the vector Laplacian (or Helmholtz operator) [S. Li, L. Lu, J.F. Wang, A graph discretization of vector Laplacian, 379 (2026) 446–460]. Motivated by the applications of graph Helmholtzian in simplicial networks, we will investiagte its basic spectral properties. As the first graph matrix indexed by edge set, we find that Helmholtzian matrix is positive semi-definite and its non-negativity correlates with the odd cycles in $G$ and the orientation on $E(G)$, while its irreducibility relates to the signed graphs with loops. We show that the eigenvalues of Helmholtzian matrix are independent of the orientation and further investigate the eigenvalue interlacing under edge additions. One of striking findings is that the non-zero eigenvalues of the Laplacian matrix are those of Helmholtzian matrix of every graph. All these discoveries reveal that the Helmholtzian spectrum of $G$ balances and bridges the oriented graphs, weighted graphs and signed graphs as well as their adjacency or Laplacian spectra.

We study tilings of rectangular boards using unit squares together with a single type of big tile shaped as a Ferrers diagram. We derive generating functions for these tilings, prove real-rootedness and interlacing properties of associated independence polynomials, and establish connections with several sequences in the OEIS. Our results touch on tilings involving L-shaped polyominoes, fault-free tilings, and cylindric variants. We prove that tiling polynomials for two-column Ferrers shapes are real-rooted and form interlacing sequences.

We investigate forbidden subgraph conditions in the complement of a graph that guarantee positive Lin–Lu–Yau curvature. In particular, we prove that every graph whose complement contains no $4$-cycles has positive Lin–Lu–Yau curvature, with the only exception of the $4$-vertex path. We further prove that, for any integer $t\ge2$, every graph on at least $\max\{t^2-2t+2, 8t\}$ vertices whose complement contains no $K_{2,t}$ has positive curvature. In addition, this lower bound on the number of vertices is optimal for $t\geq 10$. Finally, we construct examples showing that, in general, the forbidden subgraphs in these results cannot be replaced by cycles of length other than $4$ or by complete bipartite graphs $K_{s,t}$ with $s> 2$ and $t> 2$.

This paper focuses on asymptotic properties of random monomial ideals through a statistical viewpoint. It extends the study of redundancy in monomial ideals by analyzing the poset density of the LCM-lattice. We explore how this density behaves across random algebraic models and structured networks. Experimental data reveal that the LCM-lattice exhibits sharp threshold behavior rather than changing smoothly. We observe a strong negative correlation between the number of generators and LCM-lattice density, abruptly separating three distinct regimes: a low-density Taylor-like regime, a high-density redundant regime, and a narrow transition window. We show that increasing the generator degree causes this density drop to occur at lower probability thresholds. We conclude by conjecturing that for equigenerated squarefree ideals, the LCM-lattice density undergoes a sharp phase transition, analogous to the emergence of giant components in hypergraphs. This suggests that the classical, ideal-by-ideal role of the LCM-lattice as a combinatorial invariant also admits a statistical/asymptotic counterpart: in natural random families, redundancy and resolution-complexity indicators concentrate into distinct typical regimes separated by a narrow transition window.

Given a finite group $R$, we investigate the base size of the action of the automorphism group of $R$ on the lattice of subgroups of $R$. Our main result shows that this base size is $1$ if and only if $R$ is cyclic. Our motivation arises from a conjecture of Babai on the problem of representing groups as automorphism groups of lattices with a bounded number of orbits.

A perfect code in a graph $\Gamma = (V, E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent and every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A subgroup $H$ of a group $G$ is called a subgroup perfect code of $G$ if it is a perfect code in some Cayley graph of $G$. In this paper, we study the set $\Delta(G)$ of conjugacy classes of nontrivial subgroup perfect codes of $G$, with a focus on its relation to $|\pi(G)|$, the number of prime divisors of $|G|$. We prove that $|\Delta(G)| \ge |\pi(G)|$ with only three exceptional families, which leads to the natural question: when is this bound attained or nearly attained? We completely classify finite groups $G$ satisfying $|\Delta(G)| = |\pi(G)|$ and $|\Delta(G)| = |\pi(G)| + 1$, and we further characterize all insolvable groups with $|\Delta(G)| \le 6$. Our approach is based on the classification of primitive groups of odd degree, as well as the classification of primitive groups of square-free degree.

Erdős Problem 30 asks for sharp asymptotics of the Sidon extremal function $h(N)$, and Singer's construction is the classical source of lower-bound examples matching the main term. We present a Lean 4 formalization of Singer's Sidon set construction, together with reusable Sidon-set infrastructure for additive combinatorics. For every prime power $q=p^k$, we prove the existence of a Sidon set modulo $q^2+q+1$ of cardinality $q+1$; the prime-field case $q=p$ is the base presentation. The proof proceeds through a non-trivial algebraic chain: construction of the base field and its degree-three extension, analysis of the trace kernel as a 2-dimensional subspace over the base field, a geometric argument via subspace intersections establishing the multiplicative Sidon property in the quotient group, and a transfer from quotient multiplication to modular integer addition. Around this central result, we develop a reusable Sidon set library. It comprises interval and modular Sidon sets, the extremal function $h(N)$, Lindström's cross-difference inequality, a Johnson-route shift-incidence upper bound of the form $h(N)\leq\sqrt{N}+N^{1/4}+O(1)$, representation-function identities, and unconditional two-sided $h(N)=\Theta(\sqrt{N})$ bounds with exact floor-rounded finite statements for $N\geq 5$. We further formalize a conditional reduction: subpolynomial prime gaps together with a full subpolynomial upper-error hypothesis for $h(N)$ imply the Erdős Problem 30 estimate $h(N)=\sqrt{N}+O_\epsilon(N^\epsilon)$ for every $\epsilon>0$. The Singer/Sidon modules and transfer lemmas comprise 7,541 lines of Lean 4 with zero active uses of sorry. We describe the mathematical lessons learned, focusing on how formalization clarifies the precise scope of classical arguments and forces explicit treatment of the passage from the field-theoretic construction to integer Sidon predicates.

Let $G$ be a graph of order $n$ with degree sequence $d_1 \geq \cdots \geq d_{n}$. Let $m_{G}I$ be the number of signless Laplacian eigenvalues in an interval $I$. In this paper, we characterize the distribution of the signless Laplacian eigenvalues in terms of the degree sequence of a graph within specific subintervals of $[0, \, 2n-2].$ We determine all graphs $G$ such that $m_{G}[d_n, 2n-2] \leq 2, \; m_{G}[d_{n-1}, 2n-2] = 1, \; m_{G}[0, d_1] \le 2.$ We also prove that there is no graph such that $m_{G}[0, d_3]=1$. In addition, we obtain all disconnected graphs such that $m_{G}[0, d_1] = 3$. Finally, we propose two open problems for future research.

The Ramsey number $r_k(s,n)$ is the smallest integer $N$ such that every $N$-vertex $k$-graph contains either a copy of $K_s^{(k)}$ or an independent set of size $n$. A well-known conjecture of Erdős and Hajnal states that for any fixed $4\le k<s$, $r_k(s,n)\ge \operatorname{twr}_{k-1}(\Omega(n)).$ At present, only the last two cases of this conjecture remain open, namely $r_4(5,n)\ge2^{2^{\Omega(n)}}$ and $r_4(6,n)\ge2^{2^{\Omega(n)}}$. Recently, Du, Hu, Liu, and Wang achieved a breakthrough by proving $r_4(5,n)\ge 2^{2^{\Omega(n^{1/7})}}$, which is the first double-exponential lower bound for $r_4(5,n)$. In this note, we improve this to $2^{2^{\Omega(n^{1/5})}}$ by modifying their construction and reducing the greedy selection of local maxima from seven layers to five, thereby making further progress towards the Erdős-Hajnal conjecture.

An alternative generating function is proposed to enumerate row-convex polyominoes without internal holes on a discrete grid. The approach is based on integer partitions of the total area, where each partition corresponds to a sequence of row lengths, and the product of all permutations of the parts accounts for all possible horizontal alignments of consecutive rows. Summing over the products yields a formula for the total number of convex polyominoes of a given size. Numerical examples are provided for small areas, and the exact generating function is derived via a transfer series argument, establishing the asymptotic growth S(N) as A2^(N) cos(N*theta) + phi) with theta = arctan(sqrt(7)/3). The method establishes a direct connection between integer partitions and polyomino enumeration, offering a simple yet effective framework for both exact and asymptotic combinatorial analysis. Potential applications include shape priors in discrete image analysis, grid-based modeling, and combinatorial generation of convex structures.

This paper considers functional series whose terms are higher-order derivatives of Chebyshev polynomials of the second kind, where the degree of the polynomial is related to the order of the derivative. Analytic summation is used to determine the rational functions to which these series converge. These functions are expressed in terms of Chebyshev polynomials evaluated at a specific argument. Connections are established between derivatives of Chebyshev polynomials of the second kind and special numerical sequences generated by linear recurrence relations. New closed-form formulas are obtained for the sums of the series at various values of the argument. As consequences, combinatorial identities are derived for the Fibonacci, Lucas, and Pell numbers, for sections of the Fibonacci sequence, and for their convolutions. By means of analytic continuation, sums of formally divergent series are obtained, which in special cases correspond to the classical Euler formulas.

We obtain a new lower bound on the largest Sidon subset of an arbitrary finite set of integers. If $H(n)$ denotes the minimum, over all $n$-element subsets of $\mathbb Z$, of the largest Sidon subset they contain, we prove that $H(n) \geqslant \left(\frac{1}{3\sqrt 3}+o(1)\right)\sqrt n \gtrsim 0.19\sqrt n$. This improves a lower bound of Abbott related to a conjecture of Erdős on Sidon subsets of arbitrary sets of integers. The main ingredient is a compression lemma which produces, from any finite set of integers, a large subset admitting an injective Freiman $2$-morphism into a cyclic group. Combined with Singer's covering of $\mathbb Z/(q^2+q+1)\mathbb Z$ by Sidon sets, this yields the stated bound. We further extend the result to finite subsets of $\mathbb R^N$, uniformly in the dimension, by means of a projection argument and a Dirichlet approximation preserving Sidon's equation. As a consequence, every set of $n$ points in $\mathbb R^N$ contains a Sidon subset of cardinality at least $\left(\frac{1}{3\sqrt 3}+o(1)\right)\sqrt n$. We also discuss an adaptation to $B_2[g]$ sets, obtaining a lower bound of order $\frac{1}{3\sqrt 3}\sqrt{gn}$, and explain how the method can be adapted to other linear additive constraints.

For the OEIS sequence A214615, defined by $a(n) = M_{n}(1)$ where $M_{n}$ is the $n$-th Meixner polynomial satisfying $M_{n+1}(x) = x\,M_{n}(x) - n^{2}\,M_{n-1}(x)$, R.~J.~Mathar contributed on 6~March 2013 the conjectured order-2 P-recursive recurrence $a(n) - a(n-1) + (n-1)^{2}\,a(n-2) = 0$ for $n \ge 2$. We give a one-page proof. The exponential generating function $F(t) = \exp\!\bigl(\arctan t\bigr)/\sqrt{1+t^{2}}$ satisfies the first-order linear ODE $(1+t^{2})\,F'(t) = (1-t)\,F(t)$, and Mathar's recurrence then falls out by reading off the coefficient of $t^{n}/n!$. Both steps are short. The supplementary archive includes a SymPy script that checks the ODE identically and the recurrence numerically up to $n = 500$.

Coherence in a monoidal category asserts that all morphisms built from structural isomorphisms with a fixed source and target coincide. These structural isomorphisms include, in particular, the associators. Linearly distributive categories carry two tensor products, with structural morphisms given by associators and distributors relating the two tensor products. In several examples, including Grothendieck–Verdier categories, also known as $\ast$-autonomous categories, these distributors need not be invertible. We give a self-contained proof that linearly distributive categories without units are coherent, while units may obstruct coherence. With the same techniques, we also establish an analogous coherence result for Frobenius linearly distributive functors without units. These results admit a reformulation in terms of directed paths in associahedra and multiplihedra.

In this paper, we discuss combinatorial aspects of normal ordering of 3-dimensional skew polynomial rings defined and classified by Bell and Smith \cite{BellSmith1990}. With some help of the Mathematical software SageMath, we are able to reduce the length of computation of PBW forms and normal orderings appearing in commutation rules of these algebras.

In this paper, we present examples of the cyclic sieving phenomenon coming from studying independent sets in graphs of a fixed size k. Given a graph G, and a cyclic group C acting on the graph, then C also acts on the collection of independent sets of G of a fixed size k. We exhibit cyclic sieving phenomena for a cyclic group acting on the collection of independent sets of powers of cycle graphs. As a corollary, we also find a closed formula for the number of independent sets of a given size in the power of a cycle graph, and in the power of a path. We also show how the graph construction of whiskering can be used to obtain new cyclic sieving phenomena from old phenomena. We also discuss recursive techniques to exhibit cyclic sieving phenomena for the independent sets of gear graphs, helm graphs, and book graphs.

We propose a novel method to calculate logical error rates in surface codes, assuming independent and identically distributed physical errors. These results fit well known scaling laws for studied physical error rates and code distances, but break down in some other regimes. Increasing the code distance still reduces logical error rates, but returns diminish quickly beyond a certain point. We show how to use our method to analyze hypothetical quantum computers with various configurations and select designs with lower error rates. Currently, this requires expensive classical simulations of quantum decoders for various distances and physical error rates or inaccurate extrapolation from minimal experimental data. Instead, we use the symmetry of the problem to count the configurations that result in a logical error with our novel software. Given a physical error rate, we can deduce the probability of a logical error to provably good accuracy quickly. We include an analysis of measurement errors to allow for a more complete comparison of different surface code implementations. We extend this counting method to a particular locally-correlated error model and another globally-correlated error model.

We show that finitely presented groups which admit $k$-planar Cayley graphs contain finite-index subgroups with planar Cayley graphs. More generally, we answer a question of Georgakopoulos and Papasoglu in the special case of coarsely simply connected graphs: a $k$-planar, coarsely simply connected, connected, locally finite, quasi-transitive graph is quasi-isometric to a planar graph.

This paper introduces gluing diagrams, a combinatorial tool to construct homomorphisms between the shift pseudogroups of directed graphs and thus also their full groups of shifts. We will establish which of these diagrams produce isomorphisms. As an application, using the interpretation of Higman-Thompson groups as full groups of shifts of specific graphs, we will describe a procedure that constructs gluing diagrams that explicitly describe the isomorphisms between Higman-Thompson groups, conjectured by Higman and whose existence was proven by Pardo arXiv:1006.1759. This procedure will follow Euclid's algorithm and will allow us to lift isomorphisms to higher arity.