arXiv subjects

Gennian Ge

Publications and source records attributed to Gennian Ge.

At least 37 records · Page 2Linked to original sources

arXiv2026

The Frankl-Pach upper bound is not tight for any uniformity

For any positive integers $n\ge d+1\ge 3$, 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 $\binom{n}{d}$ 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 $d\ge 2$ and $n\ge 2d+2$, which demonstrates that the long-standing Frankl-Pach upper bound $\binom{n}{d}$ is not tight for any uniformity. Our proof combines a simple yet powerful polynomial method and structural analysis.

math.CO↗

arXiv2026

Universal Fourier-inertia bounds for prescribed even distances

The study of set families with restricted Hamming distances is a classical topic of extremal combinatorics and coding theory. Let $H=\{A\subseteq[n]: |A|\text{ is even}\}$ be the even subcube. Let $\ell_1,\dots,\ell_t$ be distinct positive integers and set $\mathcal L=\{2\ell_1,\dots,2\ell_t\}$. We prove that, for all sufficiently large $n$, every family $\mathcal F\subseteq H$ satisfying $ |A\triangle B|\in\mathcal L $ for all $A\ne B\in\mathcal F$ has \[ |\mathcal F|\le \sum_{i=0}^{t}\binom{n-1}{i}. \] This is best possible as a universal bound, with equality attained at the distance set $\mathcal L=\{2,4,\dots,2t\}$. 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 $\{2,4,\dots,2t\}$ among all fixed $t$-distance sets.

math.CO↗

arXiv2026

Additive structures imply more distances in $\mathbb{F}_q^d$

For a set $E \subseteq \mathbb{F}_q^d$, the distance set is defined as $Δ(E) := \{\|\mathbf{x} - \mathbf{y}\| : \mathbf{x}, \mathbf{y} \in E\}$, where $\|\cdot\|$ denotes the standard quadratic form. We investigate the Erdős--Falconer distance problem within the flexible class of $(u, s)$--Salem sets introduced by Jonathan M. Fraser, with emphasis on the even case $u = 4$. By exploiting the exact identity between $\|\widehat{E}\|_4$ and the fourth additive energy $Λ_4(E)$, we prove that quantitative gains in $Λ_4(E)$ force the existence of many distances. In particular, for a $(4, s)$--Salem set $E\subset \mathbb{F}_q^d$ with $d \geq 2$, if \[ |E|\gg q^{\min\left\{\frac{d+2}{4s+1}, \frac{d+4}{8s}\right\}}, \] then $E$ determines a positive proportion of all distances. This strictly improves Fraser's threshold of $\frac{d}{4s}$ and the Iosevich-Rudnev bound of $q^{\frac{d+1}{2}}$ in certain parameter ranges. As applications, we obtain improved thresholds for multiplicative subgroups and sets on arbitrary varieties, and establish a sharp incidence bound for Salem sets that is of independent interest in incidence geometry. Moreover, our methods give sharp lower bounds for the number of distinct distances determined by two different sets. We also propose a unified conjecture for $(4, s)$--Salem sets that reconciles known bounds and pinpoints the odd-dimensional sphere regime: in odd dimensions $d \geq 3$, the often-cited $\frac{d-1}{2}$ threshold does not follow without additional structures. This provides a clear picture of the spherical distance conjecture.

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arXiv2026

Monochromatic unit equilateral triangle on low-dimensional spheres

A result of Matoušek and Rödl in 1995 states that for every $\varepsilon>0$ and every triangle $T$ with circumradius $ρ(T)$, there exists a dimension $n=n(\varepsilon,T)$ such that every $2$-coloring of the $n$-dimensional sphere of radius $ρ(T)+\varepsilon$, namely $\mathbb{S}^{n}(ρ(T)+\varepsilon)$, contains a monochromatic congruent copy of $T$. In this paper, we determine the exact threshold dimension for the unit equilateral triangle on the sphere $\mathbb{S}^{n}(1/\sqrt{2})$: there exists a $2$-coloring of $\mathbb{S}^{2}(1/\sqrt{2})$ with no monochromatic unit equilateral triangle, whereas every $2$-coloring of $\mathbb{S}^{3}(1/\sqrt{2})$ contains one. Along the way, we also establish several further Euclidean Ramsey-type results on low-dimensional spheres, including asymmetric and isosceles variants.

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arXiv2026

Correcting One Deletion and One Substitution with a Constant Number of Reads

In this paper, we investigate the problem of designing $(n, N; \mathcal{B})$-reconstruction codes for $N\in \{14,11,9,5\}$, where $\mathcal{B}$ is the single-deletion single-substitution ball function that maps a sequence to the set of all sequences obtainable via one deletion and one substitution. Such a code is defined by the requirement that the intersection size of any two distinct single-deletion single-substitution balls is strictly less than the given number of noisy reads $N$. Note that for any $1\le N<N'$, an $(n, N; \mathcal{B})$-reconstruction code is also an $(n, N'; \mathcal{B})$-reconstruction code. It follows that the problem of designing $(n, N; \mathcal{B})$-reconstruction codes with less redundancy becomes more challenging as $N$ decreases, particularly because the problem for $N=1$ already reduces to the coding problem of single-deletion and single-substitution correcting codes. To the best of our knowledge, most existing results focus on the case where $N$ is a linear function of $n$, while only a limited number consider constant $N$. When $N=1$, the best known $(n, 1; \mathcal{B})$-reconstruction codes (single-deletion and single-substitution correcting codes) require $(4+o(1))\log n$ redundant bits. In this work, we show that this redundancy can be reduced to $3\log n+4$ when $N=5$. As $N$ increases further to $9$ and $11$, the redundancy can be improved to $2\log n+12\log\log n+O(1)$ and $\log n +12\log \log n+O(1)$, respectively. Finally, for $N=14$, we provide a reconstruction code with $\log n+3$ bits of redundancy, which is only two bits more than the best known $(n, 18; \mathcal{B})$-reconstruction codes.

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arXiv2026

Improvement on the Erdős-Kleitman conjecture via the KKL theorem

In 1974, Erdős and Kleitman conjectured that if a family $\mathcal{F}\subseteq 2^{[n]}$ contains no matching of size $s$ and is maximal with respect to this property, then $ |\mathcal{F}|\ge \left(1-2^{-(s-1)}\right)\cdot 2^{n}. $ For decades, the best general lower bound remained the trivial $2^{n-1}$. About a decade ago, Frankl and Tokushige emphasized that obtaining a bound of the form $\left(\frac{1}{2}+\varepsilon\right)\cdot 2^n$ for some $\varepsilon>0$ is a challenging problem. A breakthrough of Bucič, Letzter, Sudakov and Tran in 2018 showed that $ |\mathcal{F}|\ge \left(1-\frac{1}{s}\right)\cdot 2^n $ via two very elegant and quite different approaches. Our main result shows that $$ |\mathcal{F}|\ge \left( 1 - \frac{1}{s + (s-2)\frac{\log n}{2\sqrt{5}n}} \right)\cdot 2^n $$ by exploiting a connection to the cornerstone result of Kahn, Kalai and Linial on influences of Boolean functions. Independently, we can also obtain a weaker improvement combining the linear algebra method with a combinatorial twist.

math.CO↗

arXiv2026

Bounds and Constructions of Codes for Ordered Composite DNA Sequences

This paper extends the foundational work of Dollma \emph{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 $(e_1,\ldots,e_k)$-composite error/deletion, $e$-composite error/deletion, and the newly introduced $t$-$(e_1,\ldots,e_t)$-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 $q$, $k$, $(e_1,\ldots,e_k)$ and $e$. In contrast, previous bounds were only established for $q=2$ and limited choices of $k$, $(e_1,\ldots,e_k)$ and $e$. For CDCCs, we generalize a known non-asymptotic upper bound for $(1,0,\ldots,0)$-CDCCs and then provide a cleaner asymptotic bound. On the constructive side, for any $q\ge2$, we propose $(1,0,\ldots,0)$-CDCCs, $1$-CDCCs and $t$-$(1,\ldots,1)$-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 $(1,0,\ldots,0)$-CECC constructed by Dollma \emph{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 $t$-$(1,\ldots,1)$-CECCs. These constructions are also systematic.

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arXiv2026

Network function computation with vector linear target function and security function

In this paper, we study the problem of securely computing a function over a network, where both the target function and the security function are vector linear. The network is modeled as a directed acyclic graph. A sink node wishes to compute a function of messages generated by multiple distributed sources, while an eavesdropper can access exactly one wiretap set from a given collection. The eavesdropper must be prevented from obtaining any information about a specified security function of the source messages. The secure computing capacity is the maximum average number of times that the target function can be securely computed with zero error at the sink node with the given collection of wiretap sets and security function for one use of the network. We establish two upper bounds on this capacity, which hold for arbitrary network topologies and for any vector linear target and security functions. These bounds generalize existing results and also lead to a new upper bound when the target function is the sum over a finite field. For the lower bound, when the target function is the sum, we extend an existing method, which transforms a non-secure network code into a secure one, to the case where the security function is vector linear. Furthermore, for a particular class of networks and a vector linear target function, we characterize the required properties of the global encoding matrix to construct a secure vector linear network code.

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arXiv2026

On the Palindromic/Reverse-Complement Duplication Correcting Codes

Motivated by applications in in-vivo DNA storage, we study codes for correcting duplications. A reverse-complement duplication of length $k$ is the insertion of the reversed and complemented copy of a substring of length $k$ adjacent to its original position, while a palindromic duplication only inserts the reversed copy without complementation. We first construct an explicit code with a single redundant symbol capable of correcting an arbitrary number of reverse-complement duplications (respectively, palindromic duplications), provided that all duplications have length $k \ge 3\lceil \log_q n \rceil$ and are disjoint. Next, we derive a Gilbert-Varshamov bound for codes that can correct a reverse-complement duplication (respectively, palindromic duplication) of arbitrary length, showing that the optimal redundancy is upper bounded by $2\log_q n + \log_q\log_q n + O(1)$. Finally, for $q \ge 4$, we present two explicit constructions of codes that can correct $t$ length-one reverse-complement duplications. The first construction achieves a redundancy of $2t\log_q n + O(\log_q\log_q n)$ with encoding complexity $O(n)$ and decoding complexity $O\big(n(\log_2 n)^4\big)$. The second construction achieves an improved redundancy of $(2t-1)\log_q n + O(\log_q\log_q n)$, but with encoding and decoding complexities of $O\big(n \cdot \mathrm{poly}(\log_2 n)\big)$.

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arXiv2025

The Hilton-Milner type results of $(k, \ell)$-sum-free sets in $\mathbb F_p^n$

For a prime $p \equiv 2 \pmod 3$, it is well known that the largest sum-free subsets of $\mathbb{F}_p^n$ have size $\frac{p+1}{3} p^{n-1}$, and the extremal sets must be a cuboid of the form $\{\frac{p+1}{3}, \frac{p+1}{3}+1, \ldots, \frac{2p-1}{3}\} \times \mathbb{F}_p^{n-1}$ up to isomorphism. Recently, Reiner and Zotova proved a Hilton--Milner type stability result showing that for large $p$, any sum-free set not contained in the extremal cuboid has size at most $\frac{p-2}{3} p^{n-1}$, and all possible structures attaining this bound were classified. In this paper, we develop a general Hilton--Milner theory for $(k,\ell)$-sum-free sets in $\mathbb{F}_p^n$ for $k > \ell \ge 1$. We determine the maximum size of such sets for all $p \equiv μ\pmod{k+\ell}$ with $2 \le μ\le k+\ell-1$, and show that the extremal configurations are precisely $\lceil (μ-1)/2 \rceil$ non-isomorphic cuboids. Beyond the extremal regime, we prove sharp Hilton--Milner type stability results showing that, for all sufficiently large $p$, a $(k,\ell)$-sum-free set not contained in any of these extremal cuboids is uniformly bounded away from the maximum by a gap of order $p^{n-1}$, and we determine the full structure of all sets achieving this second-best bound in several broad parameter ranges. In particular, when $2 \le μ\le k+\ell-3$ (which is tight), only two structural types occur for all $k+\ell \ge 5$; and when $μ= 2$ or $3$, we obtain a complete classification for all $k > \ell \ge 1$. Our arguments combine additive combinatorics and Fourier-analytic methods, and make use of recent progress toward the long-standing $3k-4$ conjecture, highlighting new connections between inverse additive number theory and extremal problems over finite vector spaces.

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arXiv2025

On hierarchical secure aggregation against relay and user collusion

Secure aggregation (SA) is fundamental to privacy preservation in federated learning (FL), enabling model aggregation while preventing disclosure of individual user updates. This paper addresses hierarchical secure aggregation (HSA) against relay and user collusion in homogeneous networks, where each user connects to $n$ relays and each relay serves $m$ users. In the two-phase communication framework, users transmit masked data to relays, which then process and forward compiled messages to the server for exact sum recovery. The primary objective is to devise a transmission scheme such that the server can finish the aggregation task, while any group of $T_h$ colluding relays and $T_u$ colluding users cannot reveal any information about the data owned by the non-colluding users. In this study, we establish fundamental limits on the communication load, defined as the ratio of transmitted information size to original data size, for each user-relay link and each relay-server link. Achievable thresholds for collusion resilience are also derived. When the number of colluding relays and users falls below certain critical thresholds, we construct communication-optimal schemes using methods from network function computation. A limitation of these schemes is their reliance on large random keys. To address this, we derive a lower bound on the required key size and prove its achievability in cyclic networks, where users are connected to relays in a cyclic wrap-around manner. By establishing a connection between HSA and network function computation, this work advances the theoretical limits of communication efficiency and information-theoretic security in secure aggregation.

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arXiv2025

A Tverberg-type problem of Kalai: Two negative answers to questions of Alon and Smorodinsky, and the power of disjointness

Let $f_r(d,s_1,\ldots,s_r)$ denote the least integer $n$ such that every $n$-point set $P\subseteq\mathbb{R}^d$ admits a partition $P=P_1\cup\cdots\cup P_r$ with the property that for any choice of $s_i$-convex sets $C_i\supseteq P_i$ $(i\in[r])$ one necessarily has $\bigcap_{i=1}^r C_i\neq\emptyset$, where an $s_i$-convex set means a union of $s_i$ convex sets. A recent breakthrough by Alon and Smorodinsky establishes a general upper bound $f_r(d,s_1,\dots,s_r) = O(dr^2\log r \prod_{i=1}^r s_i\cdot \log(\prod_{i=1}^r s_i).$ Specializing to $r=2$ resolves the problem of Kalai from the 1970s. They further singled out two particularly intriguing questions: whether $f_{2}(2,s,s)$ can be improved from $O(s^2\log s)$ to $O(s)$, and whether $f_r(d,s,\ldots,s)\le Poly(r,d,s)$. We answer both in the negative by showing the exponential lower bound $f_{r}(d,s,\ldots,s)> s^{r}$ for any $r\ge 2$, $s\ge 1$ and $d\ge 2r-2$, which matches the upper bound up to a multiplicative $\log{s}$ factor for sufficiently large $s$. Our construction combines a scalloped planar configuration with a direct product of regular $s$-gon on the high-dimensional torus $(\mathbb{S}^1)^{r-2}$. Perhaps surprisingly, if we additionally require that within each block the $s_i$ convex sets are pairwise disjoint, the picture changes markedly. Let $F_r(d,s_1,\ldots,s_r)$ denote this disjoint-union variant of the extremal function. We show: (1) $F_{2}(2,s,s)=O(s\log s)$ by connecting it to a suitable line-separating function in the plane; (2) when $s$ is large, $F_r(d,s,\ldots,s)$ can be bounded by $O_{r,d}(s^{(1-\frac{1}{2^{d}(d+1)})r+1})$ and $O_{d}(r^{3}\log r\cdot s^{2d+3})$, respectively. This builds on a novel connection between the geometric obstruction and hypergraph Turán numbers, in particular, a variant of the Erdős box problem.

math.CO↗

arXiv2025

Dot-product graphs in finite fields

In this paper, we study the dot-product graphs in $\mathbb{F}_q^d$. 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.

math.CO↗

arXiv2025

Criss-Cross Deletion Correcting Codes: Optimal Constructions with Efficient Decoders

This paper addresses fundamental challenges in two-dimensional error correction by constructing optimal codes for \emph{criss-cross deletions}. We consider an $ n \times n $ array $\boldsymbol{X}$ over a $ q $-ary alphabet $Σ_q := \{0, 1, \ldots, q-1\}$ that is subject to a \emph{$(t_r, t_c)$-criss-cross deletion}, which involves the simultaneous removal of $ t_r $ rows and $ t_c $ columns. A code $\mathcal{C} \subseteq Σ_q^{n \times n}$ is defined as a \emph{$(t_r,t_c)$-criss-cross deletion correcting code} if it can successfully correct these deletions. We derive a sphere-packing type lower bound and a Gilbert-Varshamov type upper bound on the redundancy of optimal codes. Our results indicate that the optimal redundancy for a $(t_r, t_c)$-criss-cross deletion correcting code lies between $(t_r + t_c)n\log q + (t_r + t_c)\log n + O_{q,t_r,t_c}(1)$ and $(t_r + t_c)n\log q + 2(t_r + t_c)\log n + O_{q,t_r,t_c}(1)$, where the logarithm is on base two, and $O_{q,t_r,t_c}(1)$ is a constant that depends solely on $q$, $t_r$, and $t_c$. For the case of $(1,1)$-criss-cross deletions, we propose two families of constructions that achieve $2n\log q + 2\log n + O_q(1)$ bits of redundancy. This redundancy is optimal up to an additive constant term $O_q(1)$, which depends solely on $q$. One family is designed for non-binary alphabets, while the other addresses arbitrary alphabets. For the case of $(t_r, t_c)$-criss-cross deletions, we provide a strategy to derive optimal codes when both unidirectional deletions occur consecutively. We propose decoding algorithms with a time complexity of $O(n^2)$ for our codes, which are optimal for two-dimensional scenarios.

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arXiv2025

Canonical Ramsey: triangles, rectangles and beyond

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.

math.CO↗

arXiv2025

Progress towards generalized Nash-Williams' conjecture on $K_4$-decompositions

A $K_4$-decomposition of a graph is a partition of its edges into $K_4$s. A fractional $K_4$-decomposition is an assignment of a nonnegative weight to each $K_4$ in a graph such that the sum of the weights of the $K_4$s containing any given edge is one. Formulating a nonlinear programming and reducing the number of variables slowly, we prove that every graph on $n$ vertices with minimum degree at least $\frac{31}{33}n$ has a fractional $K_4$-decomposition. This improves a result of Montgomery that the same conclusion holds for graphs with minimum degree at least $\frac{399}{400}n$. Together with a result of Barber, Kühn, Lo, and Osthus, this result implies that for all $\varepsilon> 0$, every large enough $K_4$-divisible graph on $n$ vertices with minimum degree at least $(\frac{31}{33}+\varepsilon)n$ admits a $K_4$-decomposition.

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arXiv2025

Private Information Retrieval over Graphs

The problem of PIR in graph-based replication systems has received significant attention in recent years. A systematic study was conducted by Sadeh, Gu, and Tamo, where each file is replicated across two servers and the storage topology is modeled by a graph. The PIR capacity of a graph $G$, denoted by $\mathcal{C}(G)$, is defined as the supremum of retrieval rates achievable by schemes that preserve user privacy, with the rate measured as the ratio between the file size and the total number of bits downloaded. This paper makes the following key contributions. (1) The complete graph $K_N$ has emerged as a central benchmark in the study of PIR over graphs. The asymptotic gap between the upper and lower bounds for $\mathcal{C}(K_N)$ was previously 2 and was only recently reduced to $5/3$. We shrink this gap to $1.0444$, bringing it close to resolution. More precisely, (i) Sadeh, Gu, and Tamo proved that $\mathcal{C}(K_N)\le 2/(N+1)$ and conjectured this bound to be tight. We refute this conjecture by establishing the strictly stronger bound $\mathcal{C}(K_N) \le \frac{1.3922}{N}.$ We also improve the upper bound for the balanced complete bipartite graph $\mathcal{C}(K_{N/2,N/2})$. (ii) The first lower bound on $\mathcal{C}(K_N)$ was $(1+o(1))/N$, which was recently sharpened to $(6/5+o(1))/N$. We provide explicit, systematic constructions that further improve this bound, proving $\mathcal{C}(K_N)\ge(4/3-o(1))/N,$ which in particular implies $\mathcal{C}(G) \ge (4/3-o(1))/|G|$ for every graph $G$. (2) We establish a conceptual bridge between deterministic and probabilistic PIR schemes on graphs. This connection has significant implications for reducing the required subpacketization in practical implementations and is of independent interest. We also design a general probabilistic PIR scheme that performs particularly well on sparse graphs.

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arXiv2025

Codes Correcting Two Bursts of Exactly $b$ Deletions

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 $7\log n+O\left(\log n/\log\log n\right)$ bits. In contrast, our work introduces a novel approach for constructing $q$-ary codes that attain a redundancy of at most $5\log n+O(\log\log n)$ bits for all $b>1$ and $q\ge2$. Additionally, for the case where $b=1$, we present a new construction of $q$-ary two-deletion correcting codes with a redundancy of $5\log n+O(\log\log n)$ bits, for all $q>2$.

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