论文标题
Sidon Set系统的结构
The structure of Sidon set systems
论文作者
论文摘要
一个家庭$ \ MATHCAL {F} \ ABELIAN组$ G $的子集的子集2^g $是Sidon系统，如果sumsets $ a+b $带有$ a，b \ in \ mathcal {f} $是成对的。 Cilleruelo，Serra和作者先前证明了Sidon系统的最大尺寸$ f_k（n）$由$ k $ -subset组成的第一个$ n $正整数的$ c_k n^{k-1} \ leq f_k（n）\ leq f_k（n） $ k $。我们通过证明一个基本紧密的结构结果来缩小差距，尤其是$ f_k（n）\ geq（1-o（1））\ binom {n} {k-1} $。我们还使用它来建立二项式随机家族中最大的Sidon系统$ \ binom {[n]} {k} _p $的结果。还可以获得任何固定$ H \ geq 3 $的$ h $ fold总和的扩展名。
A family $\mathcal{F}\subset 2^G$ of subsets of an abelian group $G$ is a Sidon system if the sumsets $A+B$ with $A,B\in \mathcal{F}$ are pairwise distinct. Cilleruelo, Serra and the author previously proved that the maximum size $F_k(n)$ of a Sidon system consisting of $k$-subsets of the first $n$ positive integers satisfies $C_k n^{k-1}\leq F_k(n) \leq \binom{n-1}{k-1}+n-k$ for some constant $C_k$ only depending on $k$. We close the gap by proving an essentially tight structural result that in particular implies $F_k(n)\geq (1-o(1))\binom{n}{k-1}$. We also use this to establish a result about the size of the largest Sidon system in the binomial random family $\binom{[n]}{k}_p$. Extensions to $h$-fold sumsets for any fixed $h\geq 3$ are also obtained.
