A proof of the Graham Sloane conjecture

完全解决Butler-Costello-Graham关于单色星座的猜想

Complete Resolution of the Butler-Costello-Graham Conjecture on Monochromatic Constellations

本文通过证明，对于任意星座模式Q，存在一种颜色模式，使得[n]中单色星座数量达到γn²+o(n²)，其中γ小于随机着色的系数。

关于Burr、Erdős、Graham和Sós最大反Ramsey猜想

On a maximal anti-Ramsey conjecture of Burr, Erdős, Graham, and Sós

本文证实了对于奇数环C_{2k+1}（k≥4），最大反Ramsey函数的猜想，并给出了其渐近公式。

Distinct Lengths Modular Zero-sum Subsequences: A Proof of Graham's Conjecture

Let $n$ be a positive integer and let $S$ be a sequence of $n$ integers in the interval $[0,n-1]$. If there is an $r$ such that any nonempty subsequence with sum $\equiv 0$ $\pmod n$ has length $=r,$ then $S$ has at most two distinct values. This proves a conjecture of R. L. Graham. A previous result of P. Erdős and E. Szemerédi shows the validity of this conjecture if $n$ is a large prime number.

关于Ron Graham序列的一个猜想

On a Conjecture about Ron Graham's Sequence

本文研究了Ron Graham序列的性质，提出了伪多项式时间计算其值的算法，并证明了关于该函数上界的一个22年历史的猜想。

超越校正障碍的 Graham 重排猜想

Graham's rearrangement conjecture beyond the rectification barrier

本文证明 Graham 重排猜想适用于大小不超过 $e^{(\log p)^{1/4}}$ 的集合，改进已有界，并给出涉及 dissociated 集的结构定理。

Chung-Graham-Spiro间隙集猜想的一个反例

A counterexample to the Chung-Graham-Spiro gap-set conjecture

针对Chung-Graham-Spiro间隙集猜想，研究人员通过证明9属于上整数的4步间隙集但不属于下整数的4步间隙集，得出该猜想在ℓ=4时不成立的结论。

Graham's Tree Reconstruction Conjecture and a Waring-Type Problem on Partitions

Suppose $G$ is a tree. Graham's "Tree Reconstruction Conjecture" states that $G$ is uniquely determined by the integer sequence $|G|$, $|L(G)|$, $|L(L(G))|$, $|L(L(L(G)))|$, $\ldots$, where $L(H)$ denotes the line graph of the graph $H$. Little is known about this question apart from a few simple observations. We show that the number of trees on $n$ vertices which can be distinguished by their associated integer sequences is $e^{Ω((\log n)^{3/2})}$. The proof strategy involves constructing a large collection of caterpillar graphs using partitions arising from the Prouhet-Tarry-Escott problem.

On a Conjecture of Butler and Graham

Motivated by a hat guessing problem proposed by Iwasawa \cite{Iwasawa10}, Butler and Graham \cite{Butler11} made the following conjecture on the existence of certain way of marking the {\em coordinate lines} in $[k]^n$: there exists a way to mark one point on each {\em coordinate line} in $[k]^n$, so that every point in $[k]^n$ is marked exactly $a$ or $b$ times as long as the parameters $(a,b,n,k)$ satisfies that there are non-negative integers $s$ and $t$ such that $s+t = k^n$ and $as+bt = nk^{n-1}$. In this paper we prove this conjecture for any prime number $k$. Moreover, we prove the conj…

New Computational Result on Harmonious Trees

Graham and Sloane proposed in 1980 a conjecture stating that every tree has a harmonious labelling, a graph labelling closely related to additive base. Very limited results on this conjecture are known. In this paper, we proposed a computational approach to this conjecture by checking trees with limited size. With a hybrid algorithm, we are able to show that every tree with at most 31 nodes is harmonious, extending the best previous result in this direction.

Computing Bounds on Product-Graph Pebbling Numbers

Given a distribution of pebbles to the vertices of a graph, a pebbling move removes two pebbles from a single vertex and places a single pebble on an adjacent vertex. The pebbling number $π(G)$ is the smallest number such that, for any distribution of $π(G)$ pebbles to the vertices of $G$ and choice of root vertex $r$ of $G$, there exists a sequence of pebbling moves that places a pebble on $r$. Computing $π(G)$ is provably difficult, and recent methods for bounding $π(G)$ have proved computationally intractable, even for moderately sized graphs. Graham conjectured that $π(G ~\square~ H) \leq …

三重 Schubert 演算的 Graham 正性

Graham positivity of triple Schubert calculus

本文通过建立精细化 Graham 正性定理，证明 Samuel 关于三组变量中两个双 Schubert 多项式展开系数正性的猜想，并推出 Kirillov 关于斜除以差分算子作用于 Schubert 多项式正性的猜想。

关于格雷厄姆重新排列猜想

On Graham's rearrangement conjecture

本文证明了格雷厄姆重新排列猜想在特定条件下成立，解决了素数p足够大时的猜想问题。

Farey sequence and Graham's conjectures

Let ${F}_{n}$ be the Farey sequence of order $n$. For $S \subseteq {F}_n$ we let $\mathcal{Q}(S) = \left\{x/y:x,y\in S, x\le y \, \, \textrm{and} \, \, y\neq 0\right\}$. We show that if $\mathcal{Q}(S)\subseteq F_n$, then $|S|\leq n+1$. Moreover, we prove that in any of the following cases: (1) $\mathcal{Q}(S)=F_n$; (2) $\mathcal{Q}(S)\subseteq F_n$ and $|S|=n+1$, we must have $S = \left\{0,1,\frac{1}{2},\ldots,\frac{1}{n}\right\}$ or $S=\left\{0,1,\frac{1}{n},\ldots,\frac{n-1}{n}\right\}$ except for $n=4$, where we have an additional set $\{0, 1, \frac{1}{2}, \frac{1}{3}, \frac{2}{3}\}$ for t…

格雷厄姆关于交换群中小集的猜想

Graham conjecture on small sets in abelian groups

研究了交换群中大小为20以内的子集序列性，证明了其可序列化，改进了零和子集的界限，并探讨了相关CMPP猜想。

Burr-Erdős-Graham-Sós 关于七圈猜想

The Burr-Erdős-Graham-Sós conjecture for the seven-cycle

该论文证明了 Burr-Erdős-Graham-Sós 猜想在七圈情形（k=3）下成立，即 f(n,⌊n²/4⌋+1,C₇)=(1/8+o(1))n²，通过加权调色板不等式及精确有理证书给出下界，并结合正则性等方法证明上界，同时提供 Lean 4 形式化验证。

Note on a Conjecture of Graham

An old conjecture of Graham stated that if $n$ is a prime and $S$ is a sequence of $n$ terms from the cyclic group $C_n$ such that all (nontrivial) zero-sum subsequences have the same length, then $S$ must contain at most two distinct terms. In 1976, Erdős and Szemeredi gave a proof of the conjecture for sufficiently large primes $n$. However, the proof was complicated enough that the details for small primes were never worked out. Both in the paper of Erdős and Szemeredi and in a later survey by Erdős and Graham, the complexity of the proof was lamented. Recently, a new proof, valid even for …

On a conjecture of Graham on the p-divisibility of central binomial coefficients

We show that for every $r \geq 1$, and all $r$ distinct (sufficiently large) primes $p_1,..., p_r > p_0(r)$, there exist infinitely many integers $n$ such that ${2n \choose n}$ is divisible by these primes to only low multiplicity. From a theorem of Kummer, an upper bound for the number of times that a prime $p_j$ can divide ${2n \choose n}$ is $1+\log n / \log p_j$; and our theorem shows that for every $\varepsilon > 0$, $r \geq 1$, and any sufficiently large primes $p_1,...,p_r > p_0(\varepsilon,r)$, we can find integers $n$ where for $j=1,...,r$, $p_j$ divides ${2n \choose n}$ with multipli…

Girth, Pebbling, and Grid Thresholds

In this note we answer a question of Hurlbert about pebbling in graphs of high girth. Specifically we show that for every g there is a Class 0 graph of girth at least g. The proof uses the so-called Erdos construction and employs a recent result proved by Czygrinow, Hurlbert, Kierstead and Trotter. We also use the Czygrinow et al. result to prove that Graham's pebbling product conjecture holds for dense graphs. Finally, we consider a generalization of Graham's conjecture to thresholds of graph sequences and find reasonably tight bounds on the pebbling threshold of the sequence of d-dimensional…

一类半直积的 Graham 重排

Graham's rearrangement for a class of semidirect products

本文将 Graham 关于 Z_p 中部分和互异排序的猜想推广到一类半直积 Z_p ⋊_φ H，在 H 为阿贝尔群且满足部分积互异排序的条件下，得到了与 Bedert–Kravitz 同型的指数上界。

通过彩虹路径实现格雷厄姆重排猜想

Towards Graham's rearrangement conjecture via rainbow paths

通过彩虹路径方法，证明了在不加限制的情况下，格雷厄姆猜想的渐近版本成立。

格雷厄姆重排猜想的Kneser化反集中与逆吸收

Kneserized Anticoncentration and Reverse Absorption for Graham's Rearrangement Conjecture

该研究建立复合循环群的Kneser化反集中估计，结合已知结果证明格雷厄姆重排猜想对该群族成立，通过逆吸收方法推广到特定形式的循环群子集。

Sparse halves in $K_4$-free graphs

A conjecture of Chung and Graham states that every $K_4$-free graph on $n$ vertices contains a vertex set of size $\lfloor n/2 \rfloor$ that spans at most $n^2/18$ edges. We make the first step toward this conjecture by showing that it holds for all regular graphs.

部分反馈下猜牌中的极值期望：Diaconis-Graham-Spiro 两个猜想的证明

Extremal expectations in card guessing with partial feedback: proofs of two conjectures of Diaconis-Graham-Spiro

本文证明 Diaconis-Graham-Spiro 关于部分反馈猜牌中最小与最大期望得分的两个猜想，给出精确界并揭示其渐近行为。

Induced subgraphs of hypercubes and a proof of the Sensitivity Conjecture

In this paper, we show that every $(2^{n-1}+1)$-vertex induced subgraph of the $n$-dimensional cube graph has maximum degree at least $\sqrt{n}$. This result is best possible, and improves a logarithmic lower bound shown by Chung, Füredi, Graham and Seymour in 1988. As a direct consequence, we prove that the sensitivity and degree of a boolean function are polynomially related, solving an outstanding foundational problem in theoretical computer science, the Sensitivity Conjecture of Nisan and Szegedy.

A Quiver Analogue of Higman's Conjecture

An unresolved conjecture by Graham Higman states that for all $n\geq 1$ the number of conjugacy classes of the group of $n \times n$ unitriangular matrices with entries in the finite field $\mathbb{F}_q$ is a polynomial in $q$. In this paper we introduce a new quiver generalization of the conjecture. Motivated by this generalization, we prove that certain operations on quivers leave the relevant counts unchanged. Based on these invariance properties, we solve the introduced conjecture for quivers containing no path of length exceeding two, providing explicit formulas.

关于 Graham 在 $\mathbb{F}_2^n$ 上的重排猜想

On Graham's rearrangement conjecture over $\mathbb{F}_2^n$

本文研究群元素子集的有效排序问题，针对G=F_2^n情形给出本质完全解答：证明存在绝对常数C，使所有大小至少为C的子集均可有效排序；证明结合Freiman–Ruzsa定理与吸收方法，并借助Cayley图的拟随机分解结构。

为获得不同部分和对小集合进行重排

Rearranging small sets for distinct partial sums

本研究针对Graham提出的有限域非零元集合存在排序使所有部分和互异的猜想，给出了集合大小不超过$\log p / \log\log p$时的简短证明。

数学家2020年关于广义斯特林序列A001711递推猜想的简短证明

A short proof of Mathar's 2020 recurrence conjecture for the generalized-Stirling sequence A001711

本文提供了一个简短的证明，验证了Mathar对广义斯特林序列A001711的递推猜想，通过谐波数闭式表达式简化了递推关系，证明了其成立。