发表机构
Emory University（埃默里大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文推广Ruzsa定理，证明在足够大的集合中可找到满足对称线性方程且具有最多不同子集和的元素，并推广到阿贝尔群、向量空间及拟阵，给出Bondy-Simonovits定理的新证明。
AI 中文摘要
1993年，Ruzsa证明了对于每个$k \geq 2$，存在常数$C$，使得每个大小至少为$C N^{1/k}$的子集$A \subseteq [N]$包含$2k$个不同的元素$a_1, \ldots, a_k, b_1, \ldots, b_k \in A$，满足$a_1 + \cdots + a_k = b_1 + \cdots + b_k$。我们通过证明这些元素$a_1, \ldots, a_k, b_1, \ldots, b_k$可以被选择为具有额外性质来加强这一结果：集合$\{a_1, \ldots, a_k, b_1, \ldots, b_k\}$具有$2^{2k}-1$个不同的子集和，唯一的巧合是$\{a_1, \ldots, a_k\}$和$\{b_1, \ldots, b_k\}$具有相同的和。我们的证明也适用于任何奇数阶$N$的有限阿贝尔群，并且它提供了一个相应的超饱和结果：每当$|A| \geq CN^{1/k}$时，至少有$\Omega(|A|^{2k}/N)$个选择$a_1, \ldots, a_k, b_1, \ldots, b_k \in A$满足这些性质。我们证明了对于偶数阶阿贝尔群的一个稍弱陈述。我们还将我们的方法应用于向量空间设置，并证明了以下关于图中偶环极值的Bondy-Simonovits定理的$\mathbb F_q$类比：任何秩为$n$、简单、$\mathbb F_q$-可表示的拟阵，如果没有大小恰好为$2k$的回路，则其大小至多为$C q^{n/k}$，其中$C$是仅依赖于$q$和$k$的常数。当$q=2$时，对于所有$k \geq 2$，这在常数$C$的意义下是最优的。我们的方法也适用于原始图设置，并给出了Bondy-Simonovits定理及其超饱和版本的新证明。
英文摘要
In 1993, Ruzsa showed that for every $k \geq 2$, there exists a constant $C$ such that every subset $A \subseteq [N]$ of size at least $C N^{1/k}$ contains $2k$ distinct elements $a_1, \ldots, a_k, b_1, \ldots, b_k \in A$ such that $a_1 + \cdots + a_k = b_1 + \cdots + b_k$. We strengthen this result by proving that the elements $a_1, \ldots, a_k, b_1, \ldots, b_k$ can be chosen to have the additional property that $\{a_1, \ldots, a_k, b_1, \ldots, b_k\}$ has $2^{2k}-1$ distinct subset sums, with the only coincidence being that $\{a_1, \ldots, a_k\}$ and $\{b_1, \ldots, b_k\}$ have the same sum. Our proof also applies to any finite Abelian group of odd order $N$, and it provides a corresponding supersaturation result: that whenever $|A| \geq CN^{1/k}$, there are at least $Ω(|A|^{2k}/N)$ choices for $a_1, \ldots, a_k, b_1, \ldots, b_k \in A$ satisfying these properties. We prove a slightly weaker statement for Abelian groups of even order. We also apply our methods to the vector space setting and prove the following $\mathbb F_q$-analogue of the Bondy-Simonovits Theorem on the extremal number of even cycles in graphs: Any rank-$n$, simple, $\mathbb F_q$-representable matroid with no circuit of size exactly $2k$ has size at most $C q^{n/k}$ for some constant $C$ depending only on $q$ and $k$. When $q=2$, this is best possible up to the constant $C$ for all $k \geq 2$. Our methods also apply to the original graph setting and give a new proof of the Bondy-Simonovits Theorem and its supersaturation version.
Comments35 pages, 3 figures
