Related papers: Minimal and maximal Numbrix puzzles
We provide an upper bound on the uniform exponent of approximation to a triple (xi, xi^2, xi^3) by rational numbers with the same denominator, valid for any transcendental real number xi. This upper bound refines a previous result of…
Consider the family of all finite graphs with maximum degree $\Delta(G)<d$ and matching number $\nu(G)<m$. In this paper we give a new proof to obtain the exact upper bound for the number of edges in such graphs and also characterize all…
We describe the resolution of singularities of a threefold which has minimal Picard number. We describe the relation between this minimal resolution and an arbitrary resolution of singularities.
In the projective space $\mathrm{PG}(N,q)$ over the Galois field of order $q$, $N\ge3$, an iterative step-by-step construction of complete caps by adding a new point on every step is considered. It is proved that uncovered points are evenly…
The success of Large Language Models (LLMs) in human-AI collaborative decision-making hinges on their ability to provide trustworthy, gradual, and tailored explanations. Solving complex puzzles, such as Sudoku, offers a canonical example of…
The minimum $k$-enclosing ball problem seeks the ball with smallest radius that contains at least~$k$ of~$m$ given points in a general $n$-dimensional Euclidean space. This problem is NP-hard. We present a branch-and-bound algorithm on the…
We determine upper and lower bounds for the minimal number of balls of a given radius needed to cover the space of schlicht functions.
Tetravex is a widely played one person computer game in which you are given $n^2$ unit tiles, each edge of which is labelled with a number. The objective is to place each tile within a $n$ by $n$ square such that all neighbouring edges are…
We study the number of degree $n$ number fields with discriminant bounded by $X$. In this article, we improve an upper bound due to Schmidt on the number of such fields that was previously the best known upper bound for $6 \leq n \leq 94$.
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)…
The m-sophistication of a finite binary string x is introduced as a generalization of some parameter in the proof that complexity of complexity is rare. A probabilistic near sufficient statistic of x is given which length is upper bounded…
Let $\|n\|$ stand for the integer complexity of the number $n$, i.e. for the least number of $1$'s needed to write $n$ using arbitrary many additions, multiplications, and parentheses. The two-sided inequality $3\log_3 n\leq\|n\|\leq…
We derive exact and sharp lower bounds for the number of monochromatic generalized Schur triples $(x,y,x+ay)$ whose entries are from the set $\{1,\dots,n\}$, subject to a coloring with two different colors. Previously, only asymptotic…
Given a set of points in the plane, a covering path is a polygonal path that visits all the points. In this paper we consider covering paths of the vertices of an n x m grid. We show that the minimal number of segments of such a path is…
We prove that every minimal brick on n vertices has at least n/9 vertices of degree at most 4.
Proving lower bounds remains the most difficult of tasks in computational complexity theory. In this paper, we show that whereas most natural NP-complete problems belong to NLIN (linear time on nondeterministic RAMs), some of them,…
Computing lower and upper bounds on the competitive ratio of online algorithms is a challenging question: For a minimization combinatorial problem, proving a competitive ratio for a given algorithm leads to an upper bound. However computing…
We consider the possible sizes of large sumfree sets contained in the discrete hypercube $\{1,...,n\}^k$, and we determine upper and lower bounds for the maximal size as $n$ becomes large. We also discuss a continuous analogue in which our…
Rigid graphs have only finitely many realizations. In the recent years significant progress was made in computing the number of such realizations. With this progress it was also possible for the first time to do computations on large sets…
A family of subsets of the set {1,2,...,n} is said to be unbalanced if the convex hull of its characteristic vectors misses the diagonal in the n-cube.The purpose of this article is to develop the combinatorics of maximal unbalanced…