In this paper, we study the dot product graphs in F-q(d). We prove that if the size of the product of two adjacent sets is large enough, then the set of dot product graphs has positive density. Our method is based on finite field Fourier analytic techniques. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Let f_r(d,s_1,…,s_r) be the least N such that every N-point set P⊆ℝ^d has an r-partition P=P_1⊔⋯⊔ P_r with the following property: whenever C_i⊇ P_i is a union of at most s_i convex sets, one has ⋂_i=1^rC_i∅. A recent breakthrough of Alon and Smorodinsky established the first effective upper bounds f_r(d,s,…,s)≤ Cdr^2s^rlog rlog(es^r) for this problem. We obtain an asymptotically sharp lower bound by proving f_r(d,s,…,s)≥ c(d-r+2)s^rlog(s+1) for every d≥ r+2, which shows that f_r(d,s,…,s)=Θ_d,r(s^rlog s) for every fixed d≥ r+2. We also prove the general lower bound f_r(d,s,…,s)>s^min{d,r}. On the other hand, we develop a local counting argument to show that f_r(d,s,…,s)≤ C_drs^rlog(ers^r) and f_r(d,s,…,s)≤ C_dr^d+2s^d+1log(ers) whenever r≥ d+1, improving the upper bound of Alon and Smorodinsky. We also study two colored analogues. The direct Bárány–Larman-type extension, in which one seeks r disjoint rainbow sets chosen from d+1 color classes, fails as soon as two convex pieces are allowed. Nevertheless, we identify the correct colored formulation and prove a complete transversal theorem with quantitative bounds, which was also independently obtained by Keller and Smorodinsky.

In a seminal work, Cheng and Xu showed that if $S$ is a square or a triangle with a certain property, then for every positive integer $r$ there exists $n_0(S)$ independent of $r$ such that every $r$-coloring of $\mathbb{E}^n$ with $n\ge n_0(S)$ contains a monochromatic or a rainbow congruent copy of $S$. Gehér, Sagdeev, and Tóth formalized this dimension independence as the canonical Ramsey property and proved it for all hypercubes, thereby covering rectangles whose squared aspect ratio $(a/b)^2$ is rational. They asked whether this property holds for all triangles and for all rectangles. (1) We resolve both questions. More precisely, for triangles we confirm the property in $\mathbb{E}^4$ by developing a novel rotation-sphereical chaining argument. For rectangles, we introduce a structural reduction to product configurations of bounded color complexity, enabling the use of the simplex Ramsey theorem together with product Ramsey theorem. (2) Beyond this, we develop a concise perturbation framework based on an iterative embedding coupled with the Frankl-Rödl simplex super-Ramsey theorem, which yields the canonical Ramsey property for a natural class of 3-dimensional simplices and also furnishes an alternative proof for triangles.

Codes correcting bursts of deletions and localized deletions have garnered significant research interest in recent years. One of the primary objectives is to construct codes with minimal redundancy. Currently, the best known constructions of q -ary codes correcting a burst of at most t deletions ( (<= t) -burst-deletion correcting codes) achieve redundancy logn+8loglogn+o(loglogn) (for any q and t ) or logn+tloglogn+O(1) (for even q ). For codes correcting single t -localized-deletion, state-of-the-art constructions attain redundancy logn+O(t(loglogn)2) (for any q and t ) or logn+2tloglogn+O(1) (for even q ). Here, n denotes the code-length, and q and t are fixed. These codes employ a position-estimation component to approximate error positions, augmented by additional constraints that enable error-correction given the information about error positions. In this work, we first generalize the t -localized-deletion model to a (t,T) -localized-deletion model and then explore code constructions for correcting such errors. We select codewords from the set of sequences whose differential sequences are strong- (& ell;,& varepsilon;) -locally-balanced. By imposing a VT-type constraint and an L1 -weight constraint on the differential sequences of codewords, we construct novel position-estimation codes. When q >= 2 and t

For fixed d≥ 2, let τ_d(n) be the minimum size of a set S⊆{0,…,n}^d such that the affine lines determined by pairs of distinct points of S cover the grid. Let σ_d(n) be the analogous minimum when every grid point must lie on the closed segment joining two distinct points of S. A celebrated result of Alon [GAFA, 1991] proved that τ_d(n) is of order between Ω_d(n^α_d) and O_d(n^α_dlog n), where α_d=d(d-1)/2d-1, and asked whether the logarithm term is necessary. We prove that c_d n^α_d_d(n)_d(n)≤ C_d n^α_d for every fixed d≥ 2, thereby resolving Alon's problem in a stronger form.

We study vector-linear function computation over three-layer networks with a fixed target function and a fixed source-access pattern. We develop a support-constrained row-space framework that represents a linear computing code by a global row space. This space must contain the target row space and be generated by rows satisfying the local support-constraints of the network. We prove that this representation is equivalent to the existence of a linear computing code. For any prescribed global row space, we give a necessary and sufficient condition for its realization and determine the minimum uniform communication load at the middle nodes. The condition is expressed in terms of the ranks of the local subspaces supported on the source-access sets. It separates the exact local realization problem from the outer problem of designing the global row space and yields a variational characterization of the linear computing capacity. We then apply the framework to MDS targets over cyclic networks. We identify when the target row space alone is sufficient and when auxiliary rows are required. We determine the capacity in the dense regime and in the sparse regime whenever the cut-set bound is integral. For the remaining sparse parameters, we give a general linear construction whose achievable rate equals the integer part of the cut-set bound.

The sequence reconstruction problem has attracted considerable attention due to its applications in DNA storage. This problem can be modeled as transmitting a sequence over several identical noisy channels, each producing a distinct output, with the goal of reconstructing the original transmitted sequence from these outputs. The central question is to determine the minimum number of channels required to guarantee unique reconstruction of the transmitted sequence. In this paper, we consider channels that introduce k occurrences of (t, s)-bursts, where a (t, s)-burst replaces a substring of length t with another sequence of length s. For given integers t ≥ 1, s ≥ 1, k ≥ 1, n ≥ (k +1)t, we determine the minimum number of channels that guarantees unique reconstruction and design an efficient algorithm to reconstruct the transmitted sequence (of length n). Moreover, we prove that if n = (k + 1)t − 1, reconstruction is impossible in the worst-case scenario.

This paper extends the foundational work of Dollma et al. on codes for ordered composite DNA sequences. We consider the general setting with an alphabet of size q and a resolution parameter k, moving beyond the binary (q=2) case primarily studied previously. We investigate error-correcting codes for substitution errors and deletion errors under several channel models, including (e_1,…,e_k)-composite error/deletion, e-composite error/deletion, and the newly introduced t-(e_1,…,e_t)-composite error/deletion model. We first establish equivalence relations among families of composite-error correcting codes (CECCs) and among families of composite-deletion correcting codes (CDCCs). This significantly reduces the number of distinct error-parameter sets that require separate analysis. We then derive novel and general upper bounds on the sizes of CECCs using refined sphere-packing arguments and probabilistic methods. These bounds together cover all values of parameters q, k, (e_1,…,e_k) and e. In contrast, previous bounds were only established for q=2 and limited choices of k, (e_1,…,e_k) and e. For CDCCs, we generalize a known non-asymptotic upper bound for (1,0,…,0)-CDCCs and then provide a cleaner asymptotic bound. On the constructive side, for any q≥2, we propose (1,0,…,0)-CDCCs, 1-CDCCs and t-(1,…,1)-CDCCs with near-optimal redundancies. These codes have efficient and systematic encoders. For substitution errors, we design the first explicit encoding and decoding algorithms for the binary (1,0,…,0)-CECC constructed by Dollma et al, and extend the approach to general q. Furthermore, we give an improved construction of binary 1-CECCs, a construction of nonbinary 1-CECCs, and a construction of t-(1,…,1)-CECCs. These constructions are also systematic.

A family of subsets F⊆([n] k) is called intersecting if any two of its members share at least one common element. For an intersecting family, a natural extremal problem is to determine its maximum size, whereas the corresponding inverse problem seeks to characterize the extremal structures and establish their stability. The celebrated Erdős-Ko-Rado theorem addresses both the direct and inverse problems and initiated the study of intersection problems in finite set systems. In this paper, we consider a quantitative intersection problem, which can be viewed as an inverse problem in the spirit of Erdős-Ko-Rado-type theorems. For F⊆([n] k) , define its total intersection number as I( F) = ∑_F_1,F_2∈ F|F_1 ∩F_2| . We ask: what is the structure of F when it maximizes the total intersection number among all families of the same size in ([n] k) ? We provide two structural characterizations of optimal families that maximize the total intersection number. As a consequence, when n is sufficiently large and F has an appropriate size, these characterizations imply that the optimal family F is t-intersecting for some t ⩾ 1. To some extent, this reveals a connection between being intersecting and maximizing the total intersection number. Moreover, we establish upper bounds on I( F) for various ranges of | F| , and determine the unique optimal structures for certain family sizes.

Let $B^d$ denote the Euclidean unit ball in $\mathbb{R}^d$ and $f(B^d)$ denote the largest constant $c$ such that every finite collection of congruent Euclidean balls contains a pairwise disjoint subcollection whose total volume is at least $c$ times the volume of the union of the original collection. The classical Vitali covering lemma gives $f(B^d)\geq3^{-d}$. In this paper, we establish two improvements. First, by a purely combinatorial argument, we prove that $$ f(B^d)\geq \frac{2}{3^d + 2^d} $$ for every integer $d \geq1$. This improves the Vitali bound by a factor tending to $2$ as d tends to infinity. Second, using a weighted hard-core model together with a weighted geometric estimate for intersections of Euclidean balls, we show that, for all sufficiently large $d$, $$ f(B^d)\geq \left( \log\frac{3}{1+\sqrt3} -O\left(\frac{\log d}{d}\right) \right)d\,3^{-d}. $$ Thus, the classical lower bound is improved by a factor of order $d$.

Motivated by sequence reconstruction and reconstruction codes, we study extremal intersections of deletion balls over a fixed q-ary alphabet. Let Σ_q^n be the set of sequences of length n over Σ_q, and let D_t(x) denote the set of all sequences obtained from x_q^n by deleting exactly t symbols. Our first result gives a finite upper bound under a lower-order deletion-correction constraint. We prove that if x,y_q^n satisfy D_s-1(x)∩ D_s-1(y)=∅, then |D_t(x)∩ D_t(y)| ≤2ssn-st-s. For binary alphabets, this strengthens a recent asymptotic upper bound of Pham, Goyal, and Kiah (2025, JCTA). We then investigate deletion-ball intersections under simultaneous constraints on run counts and lower-order deletion-ball intersections. For fixed 0<γ≤1, integers 1≤ s≤ t, and m≥1, we show that if x,y_q^n have at most γn runs and satisfy |D_s(x)∩ D_s(y)|≤ m, then |D_t(x)∩ D_t(y)|≤mγ^t-s/(t-s)!n^t-s+O_s,t,m(n^t-s-1). Moreover, the leading term can be attainable whenever m is realized by a fixed finite-length seed pair. As a consequence, we obtain a direct lifting theorem for deletion reconstruction codes, transferring reconstruction properties from radius s to larger radii t. Finally, we establish a parallel insertion theory and derive corresponding results for insertion-ball intersections and insertion reconstruction codes.

Let L be a fixed set of positive integers. A family ℱ⊆ 2^[n] is called L-differencing if | A∖ B|∈ L for every ordered pair of distinct members A,B∈ℱ. A longstanding conjecture of Frankl, proposed in 1985, asserts that every L-differencing family has size at most n|L|. We resolve this conjecture asymptotically for every fixed L, and obtain the exact answer in the only case in which the conjectured bound could be tight. (1) If L [s] and n is large, then every L-differencing family satisfies |ℱ|≤(s/s+1+o_L(1))ns. (2) If L=[s] and n≥ 2s-1, then |ℱ|≤ns, with equality only for [n]s and [n]n-s. The first result follows by reducing directed differences to restricted Hamming distances. For the exact result, we develop a new homogeneous polynomial method, which might be of independent interest.

We study Reed–Solomon codes against adversarial coordinate permutations followed by insertion-deletion (insdel) errors. It was previously shown by Con (2025) that Reed–Solomon codes can attain the exact half-Singleton bound in this setting, but only over exponentially large alphabets. We prove that, by allowing an additive εn gap from this bound, the alphabet size can be reduced to polynomial. More precisely, for fixed constants R,ε∈(0,1) satisfying 2R+ε<1 and k=Rn, a random Reed–Solomon code of length n and dimension k over an alphabet of size n^O_R,ε(1) is, with high probability, robust against arbitrary coordinate permutations followed by up to (1-ε)n-2k+1 insdel errors. We also prove a complementary alphabet-size lower bound, showing that positive-rate codes, which are robust against linearly many insdel errors in the permutation-insdel setting, require a polynomially superlinear alphabet. Finally, for the explicit two-dimensional Reed–Solomon codes constructed by Con et al. (2024) over alphabet size O(n^3), we give an average O(n)-time decoder against arbitrary coordinate permutations followed by n-3 insdel errors. Previously, an O(n)-time decoder for this code was known only for the deletion setting.

The study of set families with restricted Hamming distances is a classical topic of extremal combinatorics and coding theory. Let H={A⊆[n]: |A| is even} be the even subcube. Let ℓ_1,…,ℓ_t be distinct positive integers and set ℒ={2ℓ_1,…,2ℓ_t}. We prove that, for all sufficiently large n, every family ℱ⊆ H satisfying |A B|∈ℒ for all A B∈ℱ has |ℱ|≤∑_i=0^tn-1i. This is best possible as a universal bound, with equality attained at the distance set ℒ={2,4,…,2t}. Our proof uses a Fourier-inertia argument based on a universal low/high boundary-layer sign pattern for the Fourier transform of the distance-polynomial kernel on the even subcube: the prescribed distances enter only through lower-order Fourier terms, while the leading boundary-layer signs depend solely on t. This even-subcube result immediately yields an odd-subcube analogue and, through parity reductions, provides bounds for arbitrary distance sets. In particular, this approach recovers the classical interval bounds of Kleitman and the corresponding interval bounds of Huang–Klurman–Pohoata, while offering a direct spectral proof of the maximality of {2,4,…,2t} among all fixed t-distance sets.

In this paper, we investigate binary reconstruction codes capable of correcting one deletion and one substitution. We define the \emph{single-deletion single-substitution ball} function $ \mathcal{B} $ as a mapping from a sequence to the set of sequences that can be derived from it by performing one deletion and one substitution. A binary \emph{$(n,N;\mathcal{B})$-reconstruction code} is defined as a collection of binary sequences of length $ n $ such that the intersection size between the single-deletion single-substitution balls of any two distinct codewords is strictly less than $ N $. This property ensures that each codeword can be uniquely reconstructed from $ N $ distinct elements in its single-deletion single-substitution ball. Our main contribution is to demonstrate that when $ N $ is set to $ 4n - 8 $, $ 3n - 4 $, $2n+9$, $ n+21 $, $31$, and $7$, the redundancy of binary $(n,N;\mathcal{B})$-reconstruction codes can be $0$, $1$, $2$, $ \log\log n + 3 $, $\log n + 1 $, and $ 3\log n + 4 $, respectively, where the logarithm is on base two.

For any positive integers n≥ d+1≥ 3, what is the maximum size of a (d+1)-uniform set system in [n] with VC-dimension at most d? In 1984, Frankl and Pach initiated the study of this fundamental problem and provided an upper bound nd via an elegant algebraic proof. Surprisingly, in 2007, Mubayi and Zhao showed that when n is sufficiently large and d is a prime power, the Frankl-Pach upper bound is not tight. They also remarked that their method requires d to be a prime power, and asked for new ideas to improve the Frankl-Pach upper bound without extra assumptions on n and d. In this paper, we provide an improvement for any d≥ 2 and n≥ 2d+2, which demonstrates that the long-standing Frankl-Pach upper bound nd is not tight for any uniformity. Our proof combines a simple yet powerful polynomial method and structural analysis.

Let B(·) be an error ball function. A set of q-ary sequences of length n is referred to as an (n, q, N; B)-reconstruction code if each sequence x within this set can be uniquely reconstructed from any N distinct elements within its error ball B(x). The main objective in this area is to determine or establish bounds for the minimum redundancy of (n, q, N; B)-reconstruction codes, denoted by ρ(n, q, N; B). In this paper, we investigate reconstruction codes where the error ball corresponds to either the t-deletion ball Dt(·) or the t-insertion ball It(·). Our primary technical contributions include: • Establishing a fundamental connection between reconstruction codes for deletions and insertions. Specifically, for any positive integers n, t, q, N, any (n, q, N; It)-reconstruction code is also an (n, q, N; Dt)-reconstruction code. This leads to the inequality ρ(n, q, N; Dt) ≤ ρ(n, q, N; It). • Identifying a significant distinction between reconstruction codes for deletions and insertions when N = O(nt−1) and t ≥ 2. For deletions, we prove that there exists some constant K such that ρ(n, q, 2(q−1)t−1/qt−1(t−1)!nt−1 + Knt−2;Dt) = O(1), which disproves a conjecture posed in [1]. In contrast, for insertions, we show that there exists some constant K′ such that ρ(n, q, (q−1)t−1 (t−1)! nt−1 + K′nt−2; It) = log log n + O(1), which extends a key result from [2]. • Constructing (n, q, N; B)-reconstruction codes, where B ∈ {D2, I2}, for N ∈ {2, 3, 4, 5} and establishing respective upper bounds of 3 log n + O(log log n), 3 log n + O(1), 2 log n + O(log log n) and log n + O(log log n) on the minimum redundancy ρ(n, q, N; B). This generalizes results previously established in [3].

For the Erdős–Frankl–Pach problem on uniform set systems of bounded VC-dimension, the Ahlswede–Khachatrian/Mubayi–Zhao construction has long served as the standard lower-bound benchmark. We develop a recursive lifting method that goes beyond this benchmark in every dimension d≥3, proving that for every d≥3 and n≥ d+3, M_d(n)≥n-1d+n-4d-2+M_d-3(n-5). The proof is elementary and proceeds through explicit trace obstructions. We also record a further recursive improvement in the concluding remarks.

We prove that all nondegenerate simplices have the canonical Ramsey property, thereby resolving a central open problem in canonical Euclidean Ramsey theory and providing a canonical counterpart to the celebrated simplex Ramsey theorem of Frankl and Rödl [JAMS, 1990].

In this paper, we investigate codes designed to correct two bursts of deletions, where each burst has a length of exactly b, where b>1. The previous best construction, achieved through the syndrome compression technique, had a redundancy of at most 7log n+O(log n/loglog n) bits. In contrast, our work introduces a novel approach for constructing q-ary codes that attain a redundancy of at most 5log n+O(loglog n) bits for all b>1 and q≥2. Additionally, for the case where b=1, we present a new construction of q-ary two-deletion correcting codes with a redundancy of 5log n+O(loglog n) bits, for all q>2.