arXiv subjects
Publications and source records attributed to Gábor Hegedűs.
Let $p\geq 3$ be a prime and $n\geq 1$ be an integer. Let $K\subseteq {\mathbb{F}_p}$ denote a fixed subset with $0\in K$. Let $A\subseteq ({\mathbb{F}_p})^n$ be an arbitrary subset such that $$\{ \mathbf{a}-\mathbf{b}:~\mathbf{a},\mathbf{b}\in A,\mathbf{a}\neq \mathbf{b}\}\cap K^n=\emptyset. $$ Then we prove the exponential upper bound $$ |A|\leq ( p-|K|+ 1 )^n. $$ We use in our proof the linear algebra bound method.
Alon, Shpilka and Umans considered the following version of usual sunflower-free subset: a subset $\mbox{$\cal F$}\subseteq \{1,\ldots ,D\}^n$ for $D>2$ is sunflower-free if for every distinct triple $x,y,z\in \mbox{$\cal F$}$ there exists a coordinate $i$ where exactly two of $x_i,y_i,z_i$ are equal. Combining the polynomial method with character theory Naslund and Sawin proved that any sunflower-free set $\mbox{$\cal F$}\subseteq \{1,\ldots ,D\}^n$ has size $$ |\mbox{$\cal F$}|\leq c_D^n, $$ where $c_D=\frac{3}{2^{2/3}}(D-1)^{2/3}$. In this short note we give a new upper bound for the size of sunflower-free subsets of $\{1,\ldots ,D\}^n$. Our main result is a new upper bound for the size of sunflower-free $k$-uniform subsets. More precisely, let $k$ be an arbitrary integer. Let $\mbox{$\cal F$}$ be a sunflower-free $k$-uniform set system. Consider $M:=|\bigcup\limits_{F\in \mbox{$\cal F$}} F|. $ Then $$ |\mbox{$\cal F$}|\leq 3(\lceil\frac{2k}{3}\rceil+1)(2^{1/3}\cdot 3e)^k(\lceil\frac Mk\rceil -1)^{\lceil\frac{2k}{3}\rceil}. $$ In the proof we use Naslund and Sawin's result about sunflower-free subsets in $\{1,\ldots ,D\}^n$.
Ellenberg and Gijswijt gave recently a new exponential upper bound for the size of three-term arithmetic progression free sets in $({\mathbb Z_p})^n$, where $p$ is a prime. Petrov summarized their method and generalized their result to linear forms. In this short note we use Petrov's result to give new exponential upper bounds for the Erdős-Ginzburg-Ziv constant of finite Abelian groups of high rank. Our main results depend on a conjecture about Property D.
Let $f(k,r,s)$ stand for the least number so that if $\cal F$ is an arbitrary $k$-uniform, $L$-intersecting set system, where $|L|=s$, and $\cal F$ has more than $f(k,r,s)$ elements, then $\cal F$ contains a sunflower with $r$ petals. We give an upper bound for $f(k,3,s)$. Let $g(k,r,\ell)$ be the least number so that any $k$-uniform, $\ell$-intersecting set system of more than $g(k,r,\ell)$ sets contains a sunflower with $r$ petals. We give also an upper bound for $g(k,r,\ell)$.