关于线性方程组可移性的Green猜想的证明

有限域上线性方程组的一个消去引理

A Removal Lemma for Systems of Linear Equations over Finite Fields

我们证明了一个删除引理有限域上的线性方程组：让$X_1,...,X_m$是有限域$\F_q$的子集，让$A$是一个$(k\times m)$矩阵，系数在$\F_q$和秩$k$ ;如果线性系统$Ax=b$有$o(q^{m-k})$解决方案与$x_i\in X_i$，那么我们可以摧毁所有这些解决方案删除$o(q)$元素从每个$X_i$。这将Green [Geometric and Functional Analysis 15（2）（2005），340- 376]关于阿贝尔群中的单个线性方程的结果扩展到线性方程的系统。特别是，我们也得到了一个类似的结果，系统的方程在整数，一个结果证明了绿色。我们的证明使用了超图移除引理的着色版本。

A tight bound for Green's arithmetic triangle removal lemma in vector spaces

Let $p$ be a fixed prime. A triangle in $\mathbb{F}_p^n$ is an ordered triple $(x,y,z)$ of points satisfying $x+y+z=0$. Let $N=p^n=|\mathbb{F}_p^n|$. Green proved an arithmetic triangle removal lemma which says that for every $ε>0$ and prime $p$, there is a $δ>0$ such that if $X,Y,Z \subset \mathbb{F}_p^n$ and the number of triangles in $X \times Y \times Z$ is at most $δN^2$, then we can delete $εN$ elements from $X$, $Y$, and $Z$ and remove all triangles. Green posed the problem of improving the quantitative bounds on the arithmetic triangle removal lemma, and, in particular, asked whether a…

群的去元引理的一个组合证明

A combinatorial proof of the Removal Lemma for Groups

Green [Geometric and泛函分析15（2005），340- 376]建立了阿贝尔群的Szemerédi正则引理的一个版本，并导出了阿贝尔群的移除引理作为其推论。我们提供了另一个证明他的删除引理，使我们能够扩展其声明的所有有限群。我们还讨论了可能的扩展的删除引理方程组。

An Arithmetic Analogue of Fox's Triangle Removal Argument

We give an arithmetic version of the recent proof of the triangle removal lemma by Fox [Fox11], for the group $\mathbb{F}_2^n$. A triangle in $\mathbb{F}_2^n$ is a triple $(x,y,z)$ such that $x+y+z = 0$. The triangle removal lemma for $\mathbb{F}_2^n$ states that for every $ε> 0$ there is a $δ> 0$, such that if a subset $A$ of $\mathbb{F}_2^n$ requires the removal of at least $ε\cdot 2^n$ elements to make it triangle-free, then it must contain at least $δ\cdot 2^{2n}$ triangles. This problem was first studied by Green [Gre05] who proved a lower bound on $δ$ using an arithmetic regularity lemma…

Removal and Stability for Erdős-Ko-Rado

A $k$-uniform family of subsets of $[n]$ is intersecting if it does not contain a disjoint pair of sets. The study of intersecting families is central to extremal set theory, dating back to the seminal Erdős-Ko-Rado theorem of 1961 that bounds the size of the largest such families. A recent trend has been to investigate the structure of set families with few disjoint pairs. Friedgut and Regev proved a general removal lemma, showing that when $γn \le k \le (\tfrac12 - γ)n$, a set family with few disjoint pairs can be made intersecting by removing few sets. We provide a simple proof of a removal…

On solution-free sets of integers

Given a linear equation $\mathcal{L}$, a set $A \subseteq [n]$ is $\mathcal{L}$-free if $A$ does not contain any `non-trivial' solutions to $\mathcal{L}$. In this paper we consider the following three general questions: (i) What is the size of the largest $\mathcal{L}$-free subset of $[n]$? (ii) How many $\mathcal{L}$-free subsets of $[n]$ are there? (iii) How many maximal $\mathcal{L}$-free subsets of $[n]$ are there? We completely resolve (i) in the case when $\mathcal{L}$ is the equation $px+qy=z$ for fixed $p,q\in \mathbb N$ where $p\geq 2$. Further, up to a multiplicative constant, we ans…