Related papers: On solution-free sets of integers

Every $n$-vertex planar triangle-free graph with maximum degree at most $3$ has an independent set of size at least $\frac{3}{8}n$. This was first conjectured by Albertson, Bollob\'as and Tucker, and was later proved by Heckman and Thomas.…

We refine previous results to provide examples, and in some cases precise classifications, of extremal subsets of {1,...,n} containing no solutions to a wide class of non-invariant, homogeneous linear equations in three variables, i.e.:…

We consider the following counting problem related to the card game SET: How many $k$-element SET-free sets are there in an $n$-dimensional SET deck? Through a series of algebraic reformulations and reinterpretations, we show the answer to…

A symmetric subset of the reals is one that remains invariant under some reflection z --> c-z. We consider, for any 0 < x <= 1, the largest real number D(x) such that every subset of $[0,1]$ with measure greater than x contains a symmetric…

A subset of the integer planar grid $[N] \times [N]$ is called corner-free if it contains no triple of the form $(x,y), (x+\delta,y), (x,y+\delta)$. It is known that such a set has a vanishingly small density, but how large this density can…

We prove that for every automata algebra of exponential growth, the associated Lie algebra contains a free subalgebra. For n\geq 1, let L_{n+2} be a Lie algebra with generator set x_1,..., x_{n+2} and the following relations: for k\leq n,…

Eberhard and Pohoata conjectured that every $3$-cube-free subset of $[N]$ has size less than $2N/3+o(N)$. In this paper we show that if we replace $[N]$ with $\mathbb{Z}_N$ the upper bound of $2N/3$ holds, and the bound is tight when $N$ is…

Given a graph G, let Q(G) denote the collection of all independent (edge-free) sets of vertices in G. We consider the problem of determining the size of a largest antichain in Q(G). When G is the edge-less graph, this problem is resolved by…

Let $x$ be a positive real number, and $\mathcal{P} \subset [2,\lambda(x)]$ be a set of primes, where $\lambda(x) \in \Omega(x^\varepsilon)$ is a monotone increasing function with $\varepsilon \in (0,1)$. We examine $Q_{\mathcal{P}}(x)$,…

Given a finite set satisfying condition $\mathcal{A}$, the subset selection problem asks, how large of a subset satisfying condition $\mathcal{B}$ can we find? We make progress on three instances of subset selection problems in planar point…

In 1990, Alon and Kleitman proposed an argument for the sum-free subset problem: every set of n nonzero elements of a finite Abelian group contains a sum-free subset A of size |A|>\frac{2}{7}n. In this note, we show that the argument…

A poset is (3+1)-free if it does not contain the disjoint union of chains of length 3 and 1 as an induced subposet. These posets are the subject of the (3+1)-free conjecture of Stanley and Stembridge. Recently, Lewis and Zhang have…

Suppose that some polynomial $f$ with rational coefficients takes only natural values at natural numbers, i.e., $L=\{f(n)\mid n\in \mathbb N\}\subset\mathbb N$. We show that the base-$q$ representation of $L$ is a context-free language if…

We study the combinatorial function $L(k,q),$ the maximum number of nonzero weights a linear code of dimension $k$ over $\F_q$ can have. We determine it completely for $q=2,$ and for $k=2,$ and provide upper and lower bounds in the general…

Capsets are subsets of $\mathbb{F}_3^n$ with no three points on a line and a capset is complete if it is not a subset of a larger capset. We study some new constructions of capsets via algebraic equations over extensions of $\mathbb{F}_3$.…

In this paper, we give a lower bound for the maximum size of a $k$-colored sum-free set in $\mathbb{Z}_m^n$, where $k\geq 3$ and $m\geq 2$ are fixed and $n$ tends to infinity. If $m$ is a prime power, this lower bound matches (up to lower…

We consider sets of positive integers containing no sum of two elements in the set and also no product of two elements. We show that the upper density of such a set is strictly smaller than 1/2 and that this is best possible. Further, we…

A set A is square-difference free (henceforth SDF) if there do not exist x,y\in A, x\ne y, such that |x-y| is a square. Let sdf(n) be the size of the largest SDF subset of {1,...,n}. Ruzsa has shown that sdf(n) = \Omega(n^{0.5(1+ \log_{65}…

We prove a lemma that is useful to get upper bounds for the number of partitions without a given subsum. From this we can deduce an improved upper bound for the number of sets represented by the (unrestricted or into unequal parts)…

Suppose $a_n$ is a real, nonnegative sequence that does not increase exponentially. For any $p<1$ we contruct a Lebesgue measurable set $E \subseteq \mathbb{R}$ which has measure at least $p$ in any unit interval and which contains no…