Michiel De Smet, Andreas Weiermann
We generalise the results by Bigorajska and Kotlarski about partitioning $α$-large sets, by extending the domain up to ordinals below $\varepsilon_ω$. These results will be very useful to give a miniaturisation of the infinite Ramsey Theorem.
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Multilevel polynomial partitioning and semialgebraic hypergraphs: regularity, Turán, and Zarankiewicz results
本研究提出多层多项式划分方案，改进了半代数超图的正则引理、Turán型与Zarankiewicz型结果，并给出多个相关领域应用。
We present some complexity results concerning the problems of covering a graph with $p$ convex sets and of partitioning a graph into $p$ convex sets. The following convexities are considered: digital convexity, monophonic convexity, $P_3$-convexity, and $P_3^*$-convexity.
In this paper, we show that every planar graph without $4$-cycles and $6$-cycles has a partition of its vertex set into two sets, where one set induces a forest, and the other induces a forest with maximum degree at most $2$ (equivalently, a disjoint union of paths). Note that we can partition the vertex set of a forest into two independent sets. However, a pair of independent sets combined may not induce a forest. Thus our result extends the result of Wang and Xu (2013) stating that the vertex set of every planar graph without $4$-cycles and $6$-cycles can be partitioned into three sets, wher…
On upper bounds on the number of parts in the problem of partitioning sets into parts of smaller diameter
本文围绕经典Borsuk问题，针对将欧氏空间中直径为1的集合划分为直径严格小于给定$b∈(0,1]$的部分的问题，讨论量$χ(n,b)$，收集其已知上界信息并给出新的上界。
Measuring the expectation value of Pauli operators on prepared quantum states is a fundamental task in a multitude of quantum algorithms. Simultaneously measuring sets of operators allows for fewer measurements and an overall speedup of the measurement process. We investigate the task of partitioning a random subset of Pauli operators into simultaneously-measurable parts. Using heuristics from coloring random graphs, we give an upper bound for the expected number of parts in our partition. We go on to conjecture that allowing arbitrary Clifford operators before measurement, rather than single-…
We give a new, systematic proof for a recent result of Larry Guth and thus also extend the result to a setting with several families of varieties: For any integer $D\geq 1$ and any collection of sets $Γ_1,\ldots,Γ_j$ of low-degree $k$-dimensional varieties in $\mathbb{R}^n$ there exists a non-zero polynomial $p\in\mathbb{R}[X_1,\ldots,X_n]$ of degree at most $D$ so that each connected component of $\mathbb{R}^n{\setminus}Z(p)$ intersects $O(jD^{k-n}|Γ_i|)$ varieties of $Γ_i$, simultaneously for every $1\leq i\leq j$. For $j=1$ we recover the original result by Guth. Our proof, via an index cal…
We prove that for every graph $H$, if a graph $G$ has no (odd) $H$ minor, then its vertex set $V(G)$ can be partitioned into three sets $X_1$, $X_2$, $X_3$ such that for each~$i$, the subgraph induced on $X_i$ has no component of size larger than a function of~$H$ and the maximum degree of~$G$. This improves a previous result of Alon, Ding, Oporowski and Vertigan~(2003) stating that $V(G)$ can be partitioned into four such sets if $G$ has no $H$ minor. Our theorem generalizes a result of Esperet and Joret~(2014), who proved it for graphs embeddable on a fixed surface and asked whether it is tr…
Partitioning Complete Geometric Graphs on Dense Point Sets into Plane Subgraphs
该研究针对稠密点集（点集最大最小距离比为Θ(√n)）的完全几何图，证明存在小于1的正常数c，使得其可划分为至多cn个无交叉平面子图，解决了特定情形下的相关公开问题。
Given two sets of natural numbers $\mathcal{A}$ and $\mathcal{B}$ of natural density $1$ we prove that their product set $\mathcal{A}\cdot \mathcal{B}:=\{ab:a\in\mathcal{A},\,b\in\mathcal{B}\}$ also has natural density $1$. On the other hand, for any $\varepsilon>0$, we show there are sets $\mathcal{A}$ of density $>1-\varepsilon$ for which the product set $\mathcal{A}\cdot\mathcal{A}$ has density $<\varepsilon$. This answers two questions of Hegyvári, Hennecart and Pach.
We show that for any finite set $A$ and an arbitrary $\varepsilon>0$ there is $k=k(\varepsilon)$ such that the higher energy ${\mathsf{E}}_k(A)$ is at most $|A|^{k+\varepsilon}$ unless $A$ has a very specific structure. As an application we obtain that any finite subset $A$ of the real numbers or the prime field either contains an additive Sidon--type subset of size $|A|^{1/2+c}$ or a multiplicative Sidon--type subset of size $|A|^{1/2+c}$.
Expanding polynomials for sets with additive structure
本文研究小和集上的二元多项式展开，利用 Solymosi–Zahl 邻近性方法并纳入和集大小，得到新的展开下界及其在加性组合与不同距离问题中的应用。
In 1976, Steinberg conjectured that planar graphs without $4$-cycles and $5$-cycles are $3$-colorable. This conjecture attracted numerous researchers for about 40 years, until it was recently disproved by Cohen-Addad et al. (2017). However, coloring planar graphs with restrictions on cycle lengths is still an active area of research, and the interest in this particular graph class remains. Let $G$ be a planar graph without $4$-cycles and $5$-cycles. For integers $d_1$ and $d_2$ satisfying $d_1+d_2\geq8$ and $d_2\geq d_1\geq 2$, it is known that $V(G)$ can be partitioned into two sets $V_1$ and…
The problem of covering the ground set of two matroids by a minimum number of common independent sets is notoriously hard even in very restricted settings, i.e.\ when the goal is to decide if two common independent sets suffice or not. Nevertheless, as the problem generalizes several long-standing open questions, identifying tractable cases is of particular interest. Strongly base orderable matroids form a class for which a basis-exchange condition that is much stronger than the standard axiom is met. As a result, several problems that are open for arbitrary matroids can be solved for this cla…
Erdős, Hajnal and Szemerédi proved that any subset $G$ of vertices of a shift graph $\text{Sh}_{n}^{k}$ has the property that the independence number of the subgraph induced by $G$ satisfies $α(\text{Sh}_{n}^{k}[G])\geq \left(\frac{1}{2}-\varepsilon\right)|G|$, where $\varepsilon\to 0$ as $k\to \infty$. In this note we prove that for $k=2$ and $n \to \infty$ there are graphs $G\subseteq \binom{[n]}{2}$ with $α(\text{Sh}_{n}^{2}[G])\leq \left(\frac{1}{4}+o(1)\right)|G|$, and $\frac{1}{4}$ is best possible. We also consider a related problem for infinite shift graphs.
We prove that if $\varepsilon(m)\to 0$ arbitrarily slowly, then for almost all $m$ and any $A\subset\mathbb{Z}_m$ such that $A-A$ does not contain non-zero quadratic residues we have $|A|\leq m^{1/2-\varepsilon(m)}.$
This paper studies partitions in the space of antimonotonic boolean functions on sets of n elements. The antimonotonic functions are the antichains of the partially ordered set of subsets. We analyse and characterise a natural partial ordering on this set. We study the inter- vals according to this ordering. We show how intervals of antimonotonic functions, and a fortiori the whole space of antimonotonic functions can be partitioned as disjoint unions of certain classes of intervals. These in- tervals are uniquely determined by antimonotonic functions on smaller sets. This leads to recursive e…
We study sets of the form $A = \big\{ n \in \mathbb N \big| \lVert p(n) \rVert_{\mathbb R / \mathbb Z} \leq \varepsilon(n) \big\}$ for various real valued polynomials $p$ and decay rates $\varepsilon$. In particular, we ask when such sets are bases of finite order for the positive integers. We show that generically, $A$ is a basis of order $2$ when $\operatorname{deg} p \geq 3$, but not when $\operatorname{deg} p = 2$, although then $A + A$ still has asymptotic density $1$.
Point sets avoiding near-integer distances
研究高维空间中避免近整数距离的点集最大数量，提出线性下界和一般上界，扩展Sárközy构造并提供双射嵌入方法。
We prove results on the structure of a subset of the circle group having positive inner Haar measure and doubling constant close to the minimum. These results go toward a continuous analogue in the circle of Freiman's $3k-4$ theorem from the integer setting. An analogue of this theorem in $\mathbb{Z}_p$ has been pursued extensively, and we use some recent results in this direction. For instance, obtaining a continuous analogue of a result of Serra and Zémor, we prove that if a subset $A$ of the circle is not too large and has doubling constant at most $2+\varepsilon$ with $\varepsilon<10^{-4}$…
$B_h$-sets and perturbations in normed vector spaces
本文证明在赋范向量空间中，若一个集合的和集具有正的最小距离，则可通过任意小的扰动将其转化为 $B_h$-集。
Fitch graphs $G=(X,E)$ are digraphs that are explained by $\{\emptyset, 1\}$-edge-labeled rooted trees $T$ with leaf set $X$: there is an arc $(x,y) \in E$ if and only if the unique path in $T$ that connects the last common ancestor $\mathrm{lca}(x,y)$ of $x$ and $y$ with $y$ contains at least one edge with label "1". In practice, Fitch graphs represent xenology relations, i.e., pairs of genes $x$ and $y$ for which a horizontal gene transfer happened along the path from $\mathrm{lca}(x,y)$ to $y$. In this contribution, we generalize the concept of Fitch graphs and consider trees $T$ that are e…
On Sets of Lengths in Monoids of plus-minus weighted Zero-Sum Sequences
本文研究加法阿贝尔群上 plus-minus 加权零和序列幺半群中的长度集合，证明了当群有限时长度集合高度结构化，而当群无限时 $\mathbb N_{\ge 2}$ 的任意有限非空子集均可作为某序列的长度集合。
Gaps in Multiplicative Sidon Sets II
本文改进了乘法Sidon集最大间隙的上界，将指数从约0.47降至10/33≈0.303。
Partitioning Theorems for Sets of Semi-Pfaffian Sets, with Applications
本文提出对半皮夫金集合族的多项式划分定理，并应用该定理推导了皮夫金版本的Szemerédi-Trotter定理及皮夫金曲线的关节数量界。
Sets of nice recurrence are partition regular
该研究证明良回归集具有分划正则性，即其任意有限着色均包含单色良回归集，解决了Bergelson的长期问题。
We say that a set of points $S\subset \mathbb{R}^d$ is an $\varepsilon$-nearly $k$-distance set if there exist $1\le t_1\le \ldots\le t_k,$ such that the distance between any two distinct points in $S$ falls into $[t_1,t_1+\varepsilon]\cup\ldots\cup[t_k,t_k+\varepsilon]$. In this paper, we study the quantity $M_k(d) = \lim_{\varepsilon\to 0}\max\{|S|\ :\ S\text{ is an }\varepsilon\text{-nearly } k \text{-distance set in } \mathbb{R}^d\}$ and its relation to the classical quantity $m_k(d)$: the size of the largest $k$-distance set in $\mathbb{R}^d$. We obtain that $M_k(d) = m_k(d)$ for $k=2,3$,…
Partitioning a Planar Graph into two Triangle-Forests
本文证明任意平面图的顶点可划分为两个集合，使每个集合诱导出无长环的三角形森林，并讨论了对局部平面图的扩展。
This paper studies visibility problems in Euclidean spaces $\mathbb{R}^d$ where the obstacles are the points of infinite discrete sets $Y\subseteq\mathbb{R}^d$. A point $x\in\mathbb{R}^d$ is called $\varepsilon$-visible for $Y$ (notation: $x\in\mathbf{vis}(Y, \varepsilon))$ if there exists a ray $L\subseteq\mathbb{R}^d$ emanating from $x$ such that $||y-z||\geq\varepsilon$, for all $y\in Y\setminus\{x\}$ and $z\in L$. A point $x\in\mathbb{R}^d$ is called visible for $Y$ (notation: $x\in\mathbf{vis}(Y))$ if $x\in\mathbf{vis}(Y, \varepsilon))$, for some $\varepsilon>0$.\\ Our main result is the …
Sparse $k$-AP Covering Sets and the Arithmetic Kakeya Conjecture
该研究否定了Kiss等人关于k-AP覆盖集的猜想，证明k≥6时存在满足特定密度界的k-AP覆盖集，并将该问题与算术Kakeya猜想建立关联。
Moment obstructions and continuum-to-discrete bounds for checkerboard no-three-in-line sets
针对棋盘无三点共线集合问题，通过矩方法证明近饱和上界，并利用连续对偶证书导出渐近常数约1.5768的线性界。
For a sequence $S$ of terms from an abelian group $G$ of length $|S|$, let $Σ_n(S)$ denote the set of all elements that can be represented as the sum of terms in some $n$-term subsequence of $S$. When the subsum set is very small, $|Σ_n(S)|\leq |S|-n+1$, it is known that the terms of $S$ can be partitioned into $n$ nonempty sets $A_1,\ldots,A_n\subseteq G$ such that $Σ_n(S)=A_1+\ldots+A_n$. Moreover, if the upper bound is strict, then $|A_i\setminus Z|\leq 1$ for all $i$, where $Z=\bigcap_{i=1}^{n}(A_i+H)$ and $H=\{g\in G:\; g+Σ_n(S)=Σ_n(S)\}$ is the stabilizer of $Σ_n(S)$. This allows structu…
Pessimal Elections for Approximately Dominating Sets
该研究针对社会选择理论中的孔多塞悖论，通过GPT-5.6 Sol Ultra构造证明，选举选出的k人委员会相关结果在放宽多数阈值后可达到常数因子范围内的紧致性。
Approximately Dominating Sets in Elections
研究证明在选举中存在常数大小的近支配集，通过概率构造和最大彩票分布，实现对选举结果的近似支配。
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