Veronica Bitonti, Emma Hogan, Tommy Walker Mackay
Let $n(k_1, k_2)$ be the least integer $n$ such that there exists a graph on $n$ vertices in which every vertex is contained in both a clique of size $k_1$ and an independent set of size $k_2$. Recently, Feige and Pauzner showed that ${n(k, k) \geq 4k-O(k^\frac{2}{3})}$, and conjectured that $n(k,k)=4k-4$. We prove this conjecture, and also establish the optimal lower bound in the more general case where $k_1$ and $k_2$ are arbitrary. We further consider the generalisation of the problem to $r$-edge-coloured complete graphs in which every vertex is contained in a size-$k$ monochromatic clique of each colour, and obtain upper and lower bounds on the size of such graphs.
Ting-Wei Chao, Zichao Dong, Hong Liu, Xichao Shu, Shuaichao Wang
We study the Zarankiewicz problem for $r$-partite, $r$-uniform intersection hypergraphs arising from $r$ families of axis-parallel boxes in $\mathbb{R}^d$ with prescribed directions $F_1, \dots, F_r \subseteq \{1, \dots, d\}$. This extends the problems studied by Chan and Har-Peled on points and $d$-dimensional boxes in $\mathbb{R}^d$, corresponding to $(F_1,F_2)=(\varnothing,[d])$, as well as by Chan, Keller, and Smorodinsky on $r$ families of $d$-dimensional boxes, corresponding to $(F_1,\dots,F_r)=([d],\dots,[d])$. Our main result establishes a sharp dichotomy for the Zarankiewicz number in this setting: it is either $Θ_r(tn^{r-1})$ or at least $Ω\bigl( tn^{r-1} \cdot \frac{\log n}{\log\log n} \bigr)$, depending only on a simple set-theoretic condition on $(F_1,\dots,F_r)$, which we call $2$-coherence. Informally, $2$-coherence captures whether the configuration contains an underlying two-dimensional incidence structure, which is precisely what gives rise to the extra polylogarithmic factor. Our proof proceeds via a sequence of reductions and a geometric slicing argument that reduces the problem to planar incidence bounds.
Revathy S. Nair, Bijo S. Anand, Julliano R. Nascimento
Given a graph $G$, a subset $S \subseteq V(G)$ is \textit{cycle convex}, if for any vertex $v \in V(G) \setminus S$, the induced subgraph, $G[S \cup \{v\}]$ cannot form a cycle containing the vertex $v$. The \textit{exchange number} of $G$, denoted by $e_{cc}(G)$ is the maximum cardinality of an $\textit{$E$-independent}$ set of $G$. This paper studies the computational complexity of determining the exchange number of graphs and provides exact values for some graph classes. Given a graph $G$ and a positive integer $k$, we show that deciding whether $e_{cc}(G) \geq k$ is NP-complete even if $G$ is a $K_5$-free graph. In contrast, we characterize all $n$-vertex graphs $G$ with exchange number $n-1$ and obtain closed formulas for chordal graphs $G$ whose blocks lie in a single chain, which leads to polynomial-time algorithms for computing $e_{cc}(G)$. We also establish a lower bound for the exchange number of the Cartesian product of general graphs and by using the results of Anand et al. \cite{bijo2}, we derive an explicit formula for the exchange number of strong and lexicographic graph products.
Mattias Ehatamm, Peter Nelson, Fernanda Rivera Omana
We generalize the well-studied notion of a modular pair of a finite matroid to arbitrary families of sets in infinite matroids, and use it to develop the theory of infinite matroids in several as-yet-unexplored areas. Our results include a complete theory of single-element extensions, a description of the relationship between quotients and projections, a proof that matroids for which every flat is modular must be finitary, and two new perspectives on the infinite matroid connectivity parameter λ. In most cases, existing theory for finite matroids either fails completely or does not extend in obvious ways, and as a result we develop multiple new techniques for reasoning about infinite matroids, including establishing well-behaved infinite analogues of nullity, local connectivity and skewness. We also point to an online repository containing formalized proofs of all our results using the lean4 proof assistant
Csilla Bujtas, Magda Dettlaff, Hanna Furmanczyk, Aleksandra Laskowska
Inspired by the majority colorings and C-colorings, we introduce and study the majority C-coloring of graphs. In such a vertex coloring, every vertex shares its color with at least half of its neighbors. The maximum number of colors that can be used in a majority C-coloring of a graph $G$ is called the majority C-chromatic number and denoted by $\mc(G)$. An upper bound on $\mc(G)$ is proved in terms of the order, minimum, and maximum degree. Its sharpness is demonstrated by several results over different graph classes. In particular, $\mc(P_n^k)= \mc(C_n^k)= \lfloor n/(k+1)\rfloor$ is true for the $k$-th power of a path and a cycle if $n \ge k+1$. Further, $\mc(G) = (n-d)/3$ holds if $G$ is a $(\mbox{claw}, K_4)$-free cubic graph and contains $d$ diamonds. %claw-free cubic graph on $n \ge 6$ vertices and contains $d$ diamonds. It is further shown that the majority C-chromatic number is not monotone under edge deletion. In fact, both the lower and upper bounds are sharp in the inequality chain $\mc(G)-2 \leq \mc(G-e) \leq \mc(G) +1$. The minimum and maximum number of edges in an $n$-vertex graph $G$ with $\mc(G)=k$ are determined for every $n$ and $k$. It is also pointed out that the classical chromatic number $χ(G)$ and $\mc(G)$ are incomparable, and the difference $\mc(G)-χ(G)$ can take any positive or negative integer. On the other hand, $\mc(G)+χ(G) \leq n+1$ holds for every graph $G$ of order $n$. The decision problem of whether $\mc(G) \ge k$ holds is NP-complete for every fixed $k\ge 2$. In contrast, some sufficient conditions for $\mc(G) \ge 2$ are proved, and a linear-time algorithm is presented that determines $\mc(T)$ if $T$ is a tree.
Daria Poliakova
We study cubic realizations of posets compatible with projection maps, meaning that the projection is represented by deletion of the last coordinate. For cylindrical projections, we introduce the pre-Reeb graph and the augmented pre-Reeb graph, which control compatible cubic lifts and compatible order-embedding cubic lifts, respectively. We apply this construction to the deletion towers in weak order of types \(A\) and \(B\). The pre-Reeb graphs are the \(1\)-skeleta of, respectively, cubes and certain zonotopes. In both cases, the augmented pre-Reeb graphs have reachability posets that are total orders, yielding combinatorial uniqueness of the compatible order-embedding cubic coordinates.
Swornalata Ojha, Hermie Monterde, Hiranmoy Pal
For $q\in\mathbb{R}\backslash\{0\}$, the generalized Laplacian of a graph $X$ is the matrix $\mathscr{L}=Δ+qA$, where $Δ$ is the degree matrix and $A$ is the adjacency matrix of $X$. In this paper, we investigate perfect state transfer (PST) on graphs with possible loops equipped with non-trivial involutions, where we take the generalized Laplacian matrix as the Hamiltonian of the underlying spin network. We establish an equivalence between the existence of PST between certain pair (or plus states) in such a graph and PST between vertices in a subgraph induced by the involution. This allows us to prove that for almost all simple unweighted planar graphs (resp., almost all simple unweighted trees), the assignment of loops of weight one to exactly two vertices in the graph produces PST between pair states relative to $\mathscr{L}$. We also show that a path on $n$ vertices admits PST between end vertices relative to $\mathscr{L}$ if and only if $n =2$, or $(n,q)=(3,\frac{k^2-l^2}{8l^2})$ where $k>l$ are integers with $k \not\equiv l \pmod{2}$. For cycles, we show that the addition of an extra edge does not yield PST between vertices relative to Laplacian and signless Laplacian matrices. Furthermore, we show that the addition of a few suitable edges (including loops) in complete bipartite graphs, cycles, and paths yields PST between pair states.
Meng Ji
Bipartite Ramsey numbers is the smallest size of a complete bipartite graph $K_{N,N}$ such that every edge-coloring with a given number of colors inevitably yields a monochromatic copy of a prescribed bipartite graph. While exact values have been determined for certain specific graphs, the general asymptotic behavior of these numbers in terms of structural graph parameters remains poorly understood. In this paper, we investigate structure-dependent growth phenomena in bipartite Ramsey theory. We first establish a general lower bound for the $q$-color bipartite Ramsey number $\operatorname{br}(K_{s,t};q)$. The proof employs a probabilistic construction together with an optimization over independent set densities, adapting the approach of Nikiforov and Sawin to the bipartite context. Next, for a fixed bipartite graph $G$ with $p$ vertices and $q$ edges, we prove a lower bound of the form $\operatorname{br}(G,K_{n,n}) > C \bigl(\frac{n}{\log n}\bigr)^{(q-1)/(p-2)}$. As a corollary, we show that sufficiently dense bipartite graphs fail to be bipartite Ramsey size linear. Turning to even cycles and complete bipartite graphs, we obtain an upper bound on the multicolor bipartite Ramsey number $\operatorname{br}_k(C_{2t};K_{n,n}) \le c_{t,k}\, n^2/\log^2 n$, which follows from classical estimates for Zarankiewicz numbers together with a double-counting argument. Building on this result, we further derive a refined linear upper bound of the form $\operatorname{br}(C_{2t},G) \le \frac{m}{2} + \frac{29t\sqrt{m}}{2}$, valid for any connected bipartite graph $G$ with $m$ edges and no isolated vertices.
Grant Molnar
We define a multi-scale metric $d_ρ$ on strings by aggregating angle distances between all $n$-gram count vectors with exponential weights $ρ^n$. We benchmark $d_ρ$ in DBSCAN clustering against edit and $n$-gram baselines, give a linear-time suffix-tree algorithm for evaluation, prove metric and stability properties (including robustness under tandem-repeat stutters), and characterize isometries.
Micha Christoph, Sahar Diskin, Lyuben Lichev, Benny Sudakov
We study the edge-expansion of the graph of a random $0/1$ polytope $P^d_p$, defined as the convex hull of a random subset of the points in $\{0,1\}^d$ where every point is retained independently and with probability $p$. This problem was introduced more than twenty years ago in a work of Gillmann and Kaibel, and has been extensively studied ever since. We prove that, for every fixed $\varepsilon>0$ and every $p\in(0,1-\varepsilon]$, with high probability the graph of $P^d_p$ has edge-expansion $Θ(d)$. This improves the previously best known bound due to Ferber, Krivelevich, Sales and Samotij, and verifies, in a strong form, the celebrated Mihail-Vazirani conjecture for random $0/1$ polytopes. Although the expansion factor $Θ(d)$ is typically best possible for $p\ge 1/2+\varepsilon$, we also show that the behaviour changes drastically at $p=1/2$. Namely, for every fixed $\varepsilon>0$ and every integer $k\ge 2$, if $p\le 1/2-\varepsilon$, then with high probability the graph of $P^d_p$ has edge-expansion $Ω(d^k)$. Thus, random $0/1$ polytopes exhibit an interesting phase transition at $p=1/2$.
Ji Ho Bae
Let H(k) be the smallest N such that every finite coloring of [N] contains a monochromatic or rainbow k-term arithmetic progression. Erdős and Graham asked whether $H(k)^{1/k}/k \to \infty$ (Problem #190 of the Erdős Problems database). We prove that there is an absolute constant $k_0 \ge 2$ such that for all $k \ge k_0$, \[ H(k)^{1/k}/k \ge (1/e - \varepsilon(k)) \cdot k/\log k, \qquad \varepsilon(k) = O(k^{-0.475} \log k) \to 0 \text{ as } k \to \infty; \] in particular $H(k)^{1/k}/k = Ω(k/\log k)$ and $\lim_{k\to\infty} H(k)^{1/k}/k = \infty$, resolving the positive direction of the Erdős-Graham question. The argument combines three standard ingredients -- the symmetric Lovász Local Lemma applied to the k-AP hypergraph on $[N]$, the restricted form of the Blankenship-Cummings-Taranchuk recurrence, and the Baker-Harman-Pintz prime-gap theorem -- together with the pigeonhole reduction $H(k) \ge W(k-1,k)$, and uses BHP as the only analytic black box. Previous applications of Erdős-Lovász had fixed $r$; the improvement here is that the $r^{k-1}$ base dominates once one allows the color count $r_0 = \lfloor k / \log k \rfloor$ to grow with $k$. No matching upper bound on $H(k)^{1/k}/k$ is known.
Hartosh Singh Bal
Let \(G\) be a finite simple graph and let \(T\) be its Hashimoto operator on the directed-edge space. We show that edge reversal induces a canonical symmetric/antisymmetric splitting under which \(T\) acquires an explicit \(2\times 2\) block form. The diagonal blocks are \(\tfrac12 L(G)\) and \(-\tfrac12 A(G)\), where \(L(G)\) is the line-graph adjacency and \(A(G)\) is the antisymmetric line-graph adjacency, while the off-diagonal block is the mixed incidence product \(M=|D|^\top D\). This identifies the ordinary and antisymmetric line-graph sectors as the two canonical diagonal sectors of Hashimoto theory and isolates a mixed sector linking them. A Schur-complement argument then gives a factorization \[ \det(I-wT)=\det\!\bigl(I-\tfrac w2 L(G)\bigr)\,C_G(w), \] where \(C_G(w)\) is an explicit correction determinant built from the antisymmetric and mixed sectors. We show that the trivial roots \(w=\pm1\) localize on canonical edge subspaces, and that for line-graph-cospectral pairs all remaining Ihara separation is forced into the correction sector. Although the raw mixed block \(M\) depends on edge orientation, its natural gauge-invariant shadows, including \(MM^\top\), \(M^\top M\), and \(M^\top L^kM\), define a canonical matrix package attached to the graph. In the regular case these collapse to adjacency-side data, but in the irregular case they need not. As an application, we exhibit irregular non-isomorphic graphs that are adjacency-cospectral and line-graph-cospectral yet are separated by the correction sector, and we find further examples where the gauge-invariant mixed shadows separate even when the scalar Ihara polynomial does not. This isolates new irregular edge-space invariants in Hashimoto--Ihara theory.
Kristina Ago, Gyula O. H. Katona
Let $\mathcal F\subset 2^{[n]}$ be an $s$-uniform family such that every two distinct sets have a nonempty intersection but intersect in at most $k$ elements. By the well-known Ray-Chaudhuri--Wilson theorem, since the intersections can take at most $k$ different values, we have $|\mathcal F|\leq \binom{n}{k}$. We give a stronger upper bound under our assumptions above, when $n$ is large enough compared to $s$ (and $k+1<s$): $|\mathcal F|\leq \frac{\binom{n-1}{k}}{\binom{s-1}{k}}$. Furthermore, we prove a generalization of the Erd\H os--Ko--Rado theorem for non-uniform families. Let $\mathcal F\subset \binom{[n]}{k}\cup\binom{[n]}{k+1}\cup\dots\cup\binom{[n]}{s}$, $3\leq k\leq s$, be a family such that for every two distinct sets the size of the intersection is between 1 and $k-1$ and $n$ is large enough then $|\mathcal F|\leq {n-1 \choose k-1}$.
Francis Brown
Using results of Fayers on the structure of Specht modules, we prove two different formulae for the determinant of matrices which are obtained by amalgamating the entries of two smaller matrices. In particular, this gives formulae for multivariable Vandermonde determinants as a sum of completely factorising terms, each of which is a Vandermonde determinant in fewer variables. As an application, we deduce an elementary proof of the multiplicativity of the transfinite diameter for products of compact sets.
Erik Pelttari, Selda Kücükçifçi, E. Şule Yazıcı
Heffter arrays are combinatorial structures used to construct orthogonal cyclic cycle decompositions and biembeddings of complete graphs onto surfaces. A Heffter array $H(m,n;h,k)$ is an $m \times n$ partially filled array with distinct nonzero entries from $\mathbb{Z}_{2nk+1}$ such that each row contains $h$ filled cells, each column contains $k$ filled cells, the elements in the filled cells form a half-set of $\mathbb{Z}_{2nk+1}$, and every row and column sums to zero modulo $2nk+1$. If these row and column sums equal zero over the integers, the structure is called an integer Heffter array. Furthermore, such an array is called globally simple if the partial sums of the entries in each row and column, evaluated in their natural order, are distinct modulo $2nk+1$. When $m=n$ and $h=k$, the array is square and denoted by $H(n;k)$. While the existence of globally simple square Heffter arrays has been established for several congruence classes, the cases where $k \equiv 1,2 \pmod{4}$ for $k > 10$ have remained an open problem [1]. In this work, we address this gap in the literature by explicitly constructing globally simple integer Heffter arrays $H(n;k)$ for the previously open cases where $k \equiv 1 \pmod{4}$ and $n \equiv 0,3 \pmod{4}$. Consequently, these constructions guarantee the existence of orthogonal cyclic $k$-cycle decompositions of the complete graph $K_{2nk+1}$ for these parameters. [1] J.H. Dinitz and A. Pasotti. A survey of Heffter arrays. In C.J. Colbourn, editor, New Advances in Designs, Codes and Cryptography, volume 86, pages 353-392. Springer Nature Switzerland, 2024.
Rocco Chirivì, Martina Costa Cesari, Xin Fang, Peter Littelmann
We define and study the higher rank GKZ-fans of point configurations, where the rank one cases coincide with the usual GKZ-fans. A point in a higher rank GKZ-fan is then used to construct higher rank quasi-valuations to degenerate the toric variety associated to the point configuration flatly to a reduced union of toric varieties. Such a union encodes the polytopal subdivision arising from the point in the higher rank GKZ-fan.
Lampros Gavalakis, Marcel K. Goh, Ioannis Kontoyiannis
Various lower bounds are established for the entropy of sums, products and their combinations. First, we derive a prime-field analogue of a version of the entropy power inequality established by Tao over torsion-free groups. Next, we prove an entropy sum-product statement: For independent and identically distributed random variables $X,X'$, the maximum of ${\bf H}(X+X')$ and ${\bf H}(XX')$ is bounded below by a linear combination of the entropy and the min-entropy (Rényi entropy of order~$\infty$) of $X$. This result, obtained by bounding entropies of the form ${\bf H}\bigl( X(Y+Z)\bigr)$ from above and below, is valid over arbitrary fields $F$. Over $F={\bf R}$, a slightly stronger inequality is derived. Finally, a weak version of a purely Shannon-entropic sum-product result is developed: If the entropic additive doubling of a random variable $X$ over an arbitrary field is $O(1)$, then its multiplicative doubling is at least proportional to ${\bf H}(X)$.
Naihuan Jing, Yinlong Liu, Jian Zhang
In this paper, we study the Littlewood theory associated with the quantum super immanants and supersymmetric polynomials, including both the super case and the quantum generalization. In the setting of quantum super Schur-Weyl duality between the quantum superalgebra $U_q(\mathfrak{gl}_{m|n})$ and the Iwahori-Hecke algebra $\mathcal{H}_r$ of type A, we explicitly construct basis vectors of the $(U_q(\mathfrak{gl}_{m|n}), \mathcal{H}_r)$-bimodule on the tensor product space $(\mathbb{C}^{m|n})^{\otimes r}$. Using this construction, we interpret the quantum super immanants via weight spaces of covariant tensor representations of $U_q(\mathfrak{gl}_{m|n})$.
Vitalii Koshelev, Alexey Koshka
This paper investigates several classical and novel variations of the Erdős--Szekeres problem, including multicolored point sets, convex hexagons with a given number of interior points, and polygons with constraints on edge colors. We propose a comprehensive computational framework combining combinatorial modeling within the SAT/ASP paradigms with the geometric realization of configurations. To determine point coordinates, we developed the \textbf{linear subreduction method}. The core idea consists of combining the complete logical model of the problem with a system of geometric inequalities, followed by fixing the abscissae to linearize the constraints. This approach enables a simultaneous search for a realization across the entire space of admissible abstract configurations (signotopes) rather than examining them individually, while linearization significantly accelerates the SMT solving process. Using this framework, we established new exact values for several functions; in particular, we proved $h_{nc}(4,0; 4,0)=26$: any bicolored set of 26 points in general position must contain the vertices of an empty monochromatic quadrilateral.
Md Moid Shaikh, Sourav Kanti Patra, Mukesh Kumar
Recently S. Goswami proved that whenever the set $\mathbb N$ of natural numbers is finitely colored, the set $\{a, b, ab, b(a+1)\}$ is monochromatic which also established a variant of the long-standing Hindman's conjecture, which asks for a monochromatic set of the form $\{a, b, ab, a+b\}$. Actually he disproved a conjecture proposed by J. Sahasrabudhe that $\{a, b, a(b + 1)\}$ is not partition regular. In this paper we prove that $\{a, b, ab, b(a+1)\}$ is monochromatic near zero which means for every finite coloring of a dense subsemigroups of $((0, \infty), +)$, the set $\{a, b, ab, b(a+1)\}$ is monochromatic near zero or in other words, we will get $a, b$ in a dense subsemigroups of $((0, \infty), +)$ as small as we want such that the set $\{a, b, ab, b(a+1)\}$ is monochromatic for every finite coloring of that dense subsemigroups of $((0, \infty), +)$, also we show that the pattern $x, y, x+y, \frac{y}{x}$ is partition regular near zero.