In 1964 Erd\H{o}s proved that $(1+\oh{1})) \frac{\eul \ln(2)}{4} k^2 2^{k}$ edges are sufficient to build a $k$-graph which is not two colorable. To this day, it known whether there exist such $k$-graphs with smaller number of edges. Erd\H{o}s' bound consequence the fact hypergraph $k^2/2$ vertices and $M(k)=(1+\oh{1}) randomly chosen size $k$ asymptotically almost surely Our first main result ...

Abstract Given a constant $$k>1$$ k > 1 , let Z be the family of round spheres radius $${{\,\mathrm{artanh}\,}}(k^{-1})$$ artanh ( - ) in hyp...

For $X(n)$ a Rademacher or Steinhaus random multiplicative function, we consider the polynomials $$ P_N(\theta) = \frac1{\sqrt{N}} \sum_{n\leq N} X(n) e(n\theta), and show that $2k$-th moments on unit circle \int_0^1 \big| \big|^{2k}\, d\theta tend to Gaussian in sense of mean-square convergence, uniformly for $k \ll (\log N / \log N)^{1/3}$, but contrast case i.i.d. coefficients, this behavior...

In this paper, we introduce the problem of Matroid-Constrained Vertex Cover: given a graph with weights on edges and matroid imposed vertices, our is to choose subset vertices that independent in matroid, objective maximizing total weight covered edges. This generalization much studied max $k$-vertex cover problem, which simple uniform it also special case monotone submodular function under con...

Abstract We present a modification of the Depth first search algorithm, suited for finding long induced paths. use it to give simple proofs following results. show that size-Ramsey number paths satisfies $\hat{R}_{\mathrm{ind}}(P_n)\leq 5 \cdot 10^7n$ , thus giving an explicit constant in linear bound, improving previous bound with large from regularity lemma argument by Haxell, Kohayakawa and ...

Let $P=(p_1, p_2, \dots, p_n)$ be a polygonal chain in $\mathbb{R}^d$. The stretch factor of $P$ is the ratio between total length and distance its endpoints, $\sum_{i = 1}^{n-1} |p_i p_{i+1}|/|p_1 p_n|$. For parameter $c \geq 1$, we call $c$-chain if $|p_ip_j|+|p_jp_k| \leq c|p_ip_k|$, for every triple $(i,j,k)$, $1 i<j<k n$. global property: it measures how close to straight line, involves al...

We offer a new definition of $varepsilon$-orthogonality in normed spaces, and we try to explain some properties of which. Also we introduce some types of $varepsilon$-orthogonality in an arbitrary  $C^ast$-algebra $mathcal{A}$, as a Hilbert $C^ast$-module over itself, and investigate some of its properties in such spaces. We state some results relating range-kernel orthogonality in $C^*$-algebras.

Quantitative bounds for random embeddings of $\mathbb{R}^{k}$ into Lorentz sequence spaces are given, with improved dependence on $\varepsilon$.