A subset of the $d$-dimensional lattice cube $[n]^d$ is sum-free if it contains no solution to the equation $x+y=z$. We study the total number of such subsets. For $d=1$, Cameron and Erd\H{o}s conjectured that the number of sum-free subsets of $[n]$ is $O(2^{n/2})$, and this was proved independently by Green and Sapozhenko. A recent work by Ghosal solved the case $d = 2$. In this paper, we consider all remaining dimensions and prove that for every fixed integer $d \geqslant 3$, the number of sum-free subsets of $[n]^d$ is $2^{M([n]^d) + O_d(n^{d-1})}$, where $M([n]^d)$ is the maximum possible size of a sum-free subset of $[n]^d$. This verifies a conjecture of Elsholtz and Rackham. Our proof combines the dual weights constructed by Keevash and Lim in their work for $M([n]^d)$, a one-dimensional counting estimate due to Ghosal, a bipartite swapping lemma of Zhao, and a strong fractional entropy inequality of Madiman and Tetali, and it avoids the use of the container lemma or deriving a stability theorem first.
Let $S^k_d(n)$ denote the maximum number of regular $(k-1)$-simplices spanned by $n$ points in $\mathbb{R}^d$. For all fixed $r\geq k\geq3$, the exact value of $S^k_{2r}(n)$ was recently determined by Dumitrescu and the authors for all sufficiently large $n$ when $k=3$, and conditionally on an optimization problem when...
For $2 \leq a \leq d+1$, what is the largest integer $H_{a,d} (n)$ such that every set of $n$ points in $\mathbb{R}^d$ with no $a$ points on a common $(a-2)$-flat contains a subset of $H_{a,d} (n)$ points whose determined $(a-1)$-dimensional simplices have pairwise distinct $(a-1)$-dimensional volumes? We construct $n$...
For positive integers $n$ and $d$, the $d$-dimensional grid of side length $n$ is the graph $G(n,d)$ with vertex set $[n]^d$, in which two vertices are adjacent if they differ by $1$ in exactly one coordinate and agree in all the other coordinates. We determine, for every fixed $d$, the asymptotic growth in $r$ of the...
The study of integer $I$ for which $I$ is a sum of cubes is a subject of study in the areas of Diophantine equations, perhaps this is due to the fact that Diophantine equations have direct applications in coding theory. Let $r, w, d$ and $m$ be positive integers and suppose that $d$ is a common difference between any t...
Lao Hussein Mude, Robert Muriungi Gitunga, Kinyanjui Jeremiah Ndung’u· Earthline Journal of Mathema...· 0 citations
For a positive integer $n$, let $S(n)=\{\sigma(d): d\mid n\}$ be ordered by divisibility, and let $a(n)$ be its width. We study how much of the divisor lattice of $n$ survives under the map $d\mapsto \sigma(d)$. For every fixed prime-exponent pattern, the largest possible value of $a(n)$ is the width of a corresponding...
Consider the linear form $L(x,y,z,w) = 3x+y-2z-2w$. For a positive integer $N$, denote by $r_L(N)$ the largest size of a subset of $\{1,2,\dots,N\}$ that avoids nontrivial solutions to $L = 0$. We show that $r_L(N) = \Omega(N^{0.5608687})$, improving the lower bound for Problem 16 in Green's list of open problems. Our...
Paul Hametner, J. Solymosi· 0 citations
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