A subset S of a poset P is called a k-family if S is the union of k or fewer antichains of P. Greene and Kleitman showed that the set of k-families of a poset forms a locally distributive lattice. Kung, Rota, and Yan asked whether every finite locally distributive lattice can be represented in this way. It is shown here that the answer is no.
A set B of positive integers is defined as a Sidon set if all differences b−b′ are distinct, where b,b′∈B and b≠b′. By confirming a problem posed by Lev at CANT 2004, Cilleruelo and Nathanson (2008) [2] obtained a nice result as follows: for every Sidon set B and every function ω(x)→∞, there exists a set A of positive integers such that every positive integer can be uniquely expressed as a−a′, where a,a′∈A and A(x)≥B(x/3)−ω(x) for all sufficiently large x. In this paper, we improve the coefficient from 1/3 to 1/2, that is: for every Sidon set B and every positive function ω(x)→∞, there exists a set A of positive integers such that every positive integer can be uniquely expressed as a−a′, where a,a′∈A andB(x/2)−ω(x)≤A(x)≤B(x/2)+ω(x) for all x≥1.
Generalized Cayley maps are embeddings of Cayley graphs into closed surfaces, either orientable or non-orientable, admitting a group of automorphisms that acts regularly on the vertices. When the surface is orientable and the automorphism group preserves orientation, these embeddings, known as Cayley maps, have been extensively studied. A seminal result by Conder and Tucker classifies orientably-regular Cayley maps on cyclic groups. This paper completes the classification of regular generalized Cayley maps on cyclic groups by addressing the remaining cases: non-orientable surfaces or orientable surfaces with the regular cyclic group acting in an orientation-reversing manner.
Let A be a set of nonnegative integers. We call A an asymptotic basis of order 2 if every sufficiently large integer can be expressed as the sum of two elements in A; otherwise, A is referred to as an asymptotic nonbasis of order 2. An asymptotic basis A of order 2 is minimal if the set A∖{a} is no longer an asymptotic basis of order 2 for every a∈A. Correspondingly, an asymptotic nonbasis A of order 2 is maximal if the set A∪{b} becomes an asymptotic basis of order 2 for every nonnegative integer b∉A. For a set A of integers, we define A(n)=|A∩[1,n]|. Let k be a positive integer. In this paper, we prove that there exists a maximal asymptotic nonbasis A={ai}i=1∞ satisfying n/2≤A(n)≤n/2+(k−1)/2 for any nonnegative integer n and limsupi→∞(ai+1−ai)=k with k≥2; we also prove that there exists a minimal asymptotic basis B={bi}i=1∞ satisfying n/2≤B(n)≤n/2+(k−1)/2 for any nonnegative integer n and limsupi→∞(bi+1−bi)=k with k≥3. These results answer two problems put forward by Tang and Chen.
In this paper we modify a fundamental block construction of Kharaghani and Seberry and show how to use certain circulant {−1,1}-matrices of odd order p to construct a complex Hadamard matrix of order 2p. In particular, for p=47 we use computer-aided methods to discover the necessary circulant matrices, and consequently give a construction of a complex Hadamard matrix of order 94 for the first time.
Motivated by the construction of optimal locally repairable codes, we introduce the new finite geometric concept of a local arc which is defined as a collection S of disjoint point sets Si in PG(2,q) such that Si∪Sj is an arc for any Si,Sj∈S. We focus on the upper and lower bounds on the sizes of maximum k-uniform local arcs. For q=pm with p prime, we construct k-uniform local arcs in PG(2,q) of size Ω(qd) where d is between 1.1167 and 1.25 depending only on m. For k=4, this implies the existence of optimal locally repairable codes (LRCs) with minimum distance 6, locality 3, and disjoint repair groups, whose length is superlinear in q–a significant improvement over the previously known O(q) constructions for such LRCs.
Using a constraint satisfaction formulation with rotational symmetry reduction, we exhibit configurations of 2n points on the n×n grid for every 2≤n≤60 with no three collinear. These computations resolved every previously open case through n=60. We describe the formulation, computational search, and empirical scaling of the method.
In this paper, we give a new proof of the Lemmens-Seidel conjecture on the maximum number of equiangular lines with a common angle $\arccos(1/5)$. This conjecture was previously resolved by Cao, Koolen, Lin, and Yu in 2022 through an analysis involving forbidden subgraphs for the smallest Seidel eigenvalue $-5$. Our new proof is based on bounds on eigenvalue multiplicities of graphs with degree no larger than $14$. To control the maximum degree of the graph associated with equiangular lines, we employ a recent inequality of Balla derived by matrix projection techniques. Our strategy also leads to a new proof for the classical result obtained by Lemmens and Seidel in 1973 for the case where the common angle is $\arccos(1/3)$.
A complete classification of the flag-transitive point-imprimitive symmetric 2-(v,k,λ) designs with v<100 is provided. Apart from the known examples with λ⩽10, the complementary designs of PG5(2) and PG3(4), and the 2-design S−(3) constructed by Kantor in [22], we found four non-isomorphic 2-(64,28,12) designs. They were constructed via computer as developments of (64,28,12)-difference sets by AbuGhneim in [1]. In the present paper, independently from [1], we construct two of the aforementioned four 2-designs and we prove that their full automorphism group is flag-transitive and point-imprimitive. The construction is theoretical and relies on the absolutely irreducible 8-dimensional F2-representation of PSL2(7). Our result, together with that about the flag-transitive point-primitive symmetric 2-designs with v<2500 by Braić-Golemac-Mandić-Vučičić [5], provides a complete classification of the flag-transitive symmetric 2-designs with v<100.
Talagrand's correlation inequality [25] provides quantitative lower bounds on the covariance of two increasing Boolean functions in terms of their coordinate influences, but, in general, a logarithmic loss is necessary. Motivated by a question of Kalai, Keller and Mossel [14, Problem 6.1], we identify a natural log-free regime. We prove that if two increasing Boolean functions on {0,1}n are either both submodular or both supermodular, thenE[fg]−E[f]E[g]≥14⋅∑i=1nInfi[f]Infi[g], where the constant 1/4 is optimal. We also prove a real-valued extension: for two functions with the same second-difference sign, the covariance is bounded below by the sum of products of their Level-1 Fourier coefficients. As a consequence, we verify the Friedgut–Kahn–Kalai–Keller spectral conjecture [11, Conjecture 5.8] in this structured setting. The proofs combine a heat-semigroup representation based on second-order discrete derivatives with an independent induction argument for the Boolean case.
The Berele row-insertion is a symplectic analogue of the Schensted row-insertion. In the present paper, we provide it with a representation theoretical interpretation via the quantum symmetric pairs of type AII. As applications, we lift Berele's Robinson–Schensted correspondence and Kobayashi–Matsumura's Robinson–Schensted–Knuth (RSK for short) correspondence to isomorphisms of representations over a quantum symmetric pair coideal subalgebra, and establish the dual RSK correspondence of type AII.
We prove that for some small and fixed δ>0, for any n, and for any three conjugacy classes C1,C2,C3 of G=Alt(n) of size at least |G|1−δ we have C1C2C3=G.The result provides a positive answer to Problem 20.23 of the Kourovka Notebook [13], improves theorems of Garonzi and Maróti [8] (using 4 classes) and Rodgers [23] (using larger classes), complements the known result for G a simple group of Lie type [20], [16], [7], and is tight in several senses. Furthermore, since no character theory is involved, the proof can be used in principle to build a constructive algorithm that, given g∈G, outputs ci∈Ci such that c1c2c3=g.
We investigate the second smallest unresolved feasible set of parameters of strongly regular graphs, (v,k,λ,μ)=(85,14,3,2). Using the classification of cubic graphs of small degree, we restrict possible local structure of such a graph G. After that, we exhaustively enumerate possible neighbourhoods of a maximal 3-clique of G and check them against a variety of conditions, including the combinatorial ones, coming from λ=3 and μ=2, as well as the linear algebra ones, utilising the Euclidean representation of G. These conditions yield contradiction in all cases, and hence, no srg(85,14,3,2) exists.
We construct a new family of permutation group polynomials over finite fields of arbitrary characteristic, which are special types of bivariate local permutation polynomials. For this family, we explicitly construct their companion. We also determine the total number of permutation group polynomials of this form. Moreover, we resolve the problem of enumerating e-Klenian polynomials over finite fields for e≥ 1, a problem previously noted as nontrivial by Gutierrez and Urroz (2023). In addition, we provide the exact number of permutation group polynomials equivalent to our proposed permutation group polynomials, as well as the exact number of those permutation group polynomials equivalent to e-Klenian polynomials.
We completely determine the asymptotic depth, equivalently, the asymptotic projective dimension of a chain of edge ideals that is invariant under the action of the monoid Inc of increasing functions on the positive integers. Our results and their proofs also reveal surprising combinatorial and topological properties of corresponding graphs and their independence complexes. In particular, we are able to determine the asymptotic behavior of all reduced homology groups of these independence complexes. (c) 2026 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Lattice tilings of Z(n) by limited-magnitude error balls correspond to linear perfect codes under such error models and play a crucial role in flash memory applications. In this work, we establish three main results on lattice tilings of Z(n) by limited-magnitude error balls B(n, 2, k(1), k(2)). First, we fully determine the existence of lattice tilings by B(n, 2, 3,0) in all dimensions n. Second, we completely resolve the case k(1) = k(2) + 1. Finally, we prove that for any integers k(1) > k(2) >= 0 such that k(1) + k(2) + 1 is composite, no lattice tiling of Z(n) by the error ball 13(n, 2, k(1), k(2)) exists for sufficiently large n. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Let Gamma denote a distance-regular graph with vertex set X and diameter D >= 3. Fix a vertex x is an element of X. Let the field F be either R or C. Let MatX(F) denote the F-algebra of matrices whose rows and columns are indexed by X and all entries in F. The Terwilliger algebra TF= TF (x) is the subalgebra of MatX (F) generated by the adjacency matrix A of Gamma and the dual primitive idempotents {Ei & lowast; }Di=0 of Gamma with respect to x. Let {Ei}Di=0 denote the primitive idempotents of A. Assume that the ordering {Ei}Di=0 is Q-polynomial. Let W denote an irreducible TF-module. We say that W is sharp over F whenever dim(Er & lowast; W) = 1, where r is the endpoint of W. It is known, by Nomura and Terwilliger (2008), that every irreducible TC-module is sharp. In this paper, we prove that every irreducible TR-module is sharp. Once this is established, we obtain four additional results: (i) if W is an irreducible TR-module, then its complexification WC = W circle times R Cis an irreducible TC-module; (ii) two irreducible TR-modules W1 and W2 are isomorphic if and only if their complexifications W1Cand W2C are isomorphic as TC-modules; (iii) if (R) hi=1 Matni(C) is the Wedderburn decomposition of TC, then (R) hi=1 Matni (R) is the Wedderburn decomposition of TR; (iv) each of the subalgebras E1 & lowast;TE1 & lowast;, E1TE1, E & lowast;DTE & lowast;D, and EDT ED is commutative and every element of these algebras is a symmetric matrix. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
