Related papers: Upper bounds for $B_h[g]$-sets with small $h$
The bondage number $b(G)$ of a graph $G$ is the smallest number of edges whose removal from $G$ results in a graph with larger domination number. Recently Gagarin and Zverovich showed that, for a graph $G$ with maximum degree $\Delta(G)$…
We give a very short argument showing how the main identity from our earlier paper (Sogge and Zelditch, 2011) immediately leads to the best lower bound currently known (Colding and Minicozzi II, 2011) for the Hausdorff measure of nodal sets…
In this paper, we focus on estimating measure upper bounds of nodal sets of solutions to the following boundary value problem \begin{equation*} \left\{ \begin{array}{lll} \Delta u+Vu=0\quad \mbox{in}\ \Omega,\\[2mm] u=0\quad \mbox{on}\…
It is a well-known conjecture, sometimes attributed to Frankl, that for any family of sets which is closed under the union operation, there is some element which is contained in at least half of the sets. Gilmer was the first to prove a…
Let $B$ be a set of natural numbers of size $n$. We prove that the length of the longest arithmetic progression contained in the product set $B.B = \{bb'| \, b, b' \in B\}$ cannot be greater than $O(n \log n)$ which matches the lower bound…
For fixed integers $b\geq k$, the problem of perfect $(b,k)$-hashing asks for the asymptotic growth of largest subsets of $\{1,2,\ldots,b\}^n$ such that for any $k$ distinct elements in the set, there is a coordinate where they all differ.…
In the minimum supersymmetric SU(5) GUT, we derive the upper limit on the mass of the color-triplet Higgs multiplets as $\mhc\leq 2.4\times 10^{16}~\GEV$ (90 \% C.L.) taking all possible corrections into account in a renormalization group…
We derive a new estimate of the size of finite sets of points in metric spaces with few distances. The following applications are considered: (1) we improve the Ray-Chaudhuri--Wilson bound of the size of uniform intersecting families of…
Motivated by applications to matrix multiplication algorithms, Pratt asked (ITCS'24) how large a subset of $[n] \times [n]$ could be without containing a skew-corner: three points $(x,y), (x,y+h),(x+h,y')$ with $h \ne 0$. We prove any skew…
For $b\in\mathbb{N}, b\ge2$ we determine the limit points of certain subsets of $$ \left\{\frac{b^n\pmod{n}}{n}:n\in\mathbb{N}\right\}. $$ As a consequence, we obtain the density of the latter set in $[0,1]$, a result first established in…
Cameron and Erd\H{o}s asked whether the number of \emph{maximal} sum-free sets in $\{1, \dots , n\}$ is much smaller than the number of sum-free sets. In the same paper they gave a lower bound of $2^{\lfloor n/4 \rfloor }$ for the number of…
Let $S_g$ be a closed orientable surface of genus $g\geq 2$. A collection $\Omega = \{ \gamma_1, \dots, \gamma_s\}$ of pairwise non-homotopic simple closed curves on $S_g$ such that $\gamma_i$ and $\gamma_j$ are in minimal position, is…
Dorais asked for the maximum guaranteed size of a dimension $d$ subposet of an $n$-element poset. A lower bound of order $\sqrt{n}$ was found by Goodwillie. We provide a sublinear upper bound for each $d$. For $d=2$, our bound is…
The minimal number of 3He atoms required to form a bound cluster has been estimated by means of a Diffusion Monte Carlo procedure within the fixed-node approximation. Several importance sampling wave functions have been employed in order to…
We estimate the upper bound of the Higgs mass and allowed parameter region in the anomaly mediated supersymmetry breaking models. There has been a difficulty that the parameter region cannot be determined precisely in almost SUSY models.…
We derive here a robust bound on the effective number of neutrinos from constraints on primordial nucleosynthesis yields of deuterium and helium. In particular, our results are based on very weak assumptions on the astrophysical…
For $h \geq 1$, a $B_h$-set is a set of integers such that every integer $n$ has at most one representation in the form $n = a_{i_1} + \cdots + a_{i_h}$, where $a_{i_r} \in A$ for all $r = 1,\ldots, h$ and $a_{i_1} \leq \ldots \leq…
We study solutions to the equation $a+b=c$, where $a,b,c$ form a triple of coprime natural numbers. The $abc$ conjecture asserts that, for any $\epsilon>0$, such triples satisfy $\mathrm{rad}(abc) \ge c^{1-\epsilon}$ with finitely many…
A subset $S$ of real numbers is called bi-Sidon if it is a Sidon set with respect to both addition and multiplication, i.e., if all pairwise sums and all pairwise products of elements of $S$ are distinct. Imre Ruzsa asked the following…
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…