Related papers: Improved Bounds for Progression-Free Sets in $C_{8…

For a family $\mathcal{F}$ of graphs, let $ex(n,\mathcal{F})$ denote the maximum number of edges in an $n$-vertex graph which contains none of the members of $\mathcal{F}$ as a subgraph. A longstanding problem in extremal graph theory asks…

We give an independent set of size $367$ in the fifth strong product power of $C_7$, where $C_7$ is the cycle on $7$ vertices. This leads to an improved lower bound on the Shannon capacity of $C_7$: $\Theta(C_7)\geq 367^{1/5} > 3.2578$. The…

Let $G_{(m,3,r)}=\langle x,y\mid x^m=1, y^3=1,yx=x^ry\rangle$ be a metacyclic group of order $3m$, where ${\rm gcd}(m,r)=1$, $1<r<m$ and $r^3\equiv 1$ (mod $m$). Then left ideals of the group algebra $\mathbb{F}_q[G_{(m,3,r)}]$ are called…

Green and Sisask showed that the maximal number of $3$-term arithmetic progressions in $n$-element sets of integers is $\lceil n^2/2\rceil$; it is easy to see that the same holds if the set of integers is replaced by the real line or by any…

The Gowers U^3 norm is one of a sequence of norms used in the study of arithmetic progressions. If G is an abelian group and A is a subset of G then the U^3(G) of the characteristic function 1_A is useful in the study of progressions of…

Let $q$ be an odd prime power. Combining the discussion of Varnavides and a recent theorem of Ellenberg and Gijswijt, we show that a subset $A\subset{\mathbb F}_q^n$ will contain many non-trivial three-term arithmetic progressions, whenever…

Let $G$ be a finite group, define $I(G)=\{x\in G : x^{2}=1\}$, $C(G)=$ set of the cyclic subgroups of $G$, $i(G)=|I(G)|$ and $c(G)=|C(G)|$. In this article, we will classify finite groups with $i(G)=c(G)-r$ for $r=0,1,$ and $2$. We also…

In 1980 Rostislav Grigorchuk constructed a group $G$ of intermediate growth, and later obtained the following estimates on its growth $\gamma$: $e^{\sqrt{n}}\precsim\gamma(n)\precsim e^{n^\beta},$ where $\beta=\log_{32}(31)\approx0.991$. He…

Let $G$ be a connected graph on $n$ vertices. The Gallai number $Gal(G)$ of $G$ is the size of the smallest set of vertices that meets every maximum path in $G$. Gr\"unbaum constructed a graph $G$ with $Gal(G)=3$. Very recently, Long,…

Let G be a finitely connected Lie group and let K be a maximal compact subgroup. Let M be a cocompact G-proper manifold with boundary, endowed with a G-invariant metric which is of product type near the boundary. Under additional…

For a finite abelian group $G$ and a positive integer $h$, the unrestricted (resp.~restricted) $h$-critical number $\chi(G,h)$ (resp.~$\chi \hat{\;}(G,h)$) of $G$ is defined to be the minimum value of $m$, if exists, for which the $h$-fold…

Let G denote a closed, connected, self adjoint, noncompact subgroup of GL(n,R), and let d_{R} denote the canonical right invariant Riemannian metric on G. For v in R^{n} let G_{v} = {g in G : g(v) = v}. We obtain algebraically defined upper…

We say that a set is a multiplicative 3-Sidon set if the equation $s_1s_2s_3=t_1t_2t_3$ does not have a solution consisting of distinct elements taken from this set. In this paper we show that the size of a multiplicative 3-Sidon subset of…

For a positive integer $r$, let $G(r)$ be the smallest $N$ such that, whenever the edges of the Cartesian product $K_N \times K_N$ are $r$-coloured, then there is a rectangle in which both pairs of opposite edges receive the same colour. In…

Recently, Croot, Lev, and Pach (Ann. of Math., 185:331--337, 2017.) and Ellenberg and Gijswijt (Ann. of Math., 185:339--443, 2017.) developed a new polynomial method and used it to prove upper bounds for three-term arithmetic progression…

We consider the periods of the linear congruential and the power generators modulo $n$ and, for fixed choices of initial parameters, give lower bounds that hold for ``most'' $n$ when $n$ ranges over three different sets: the set of primes,…

In 1979, Erd\H{o}s conjectured that if $m = O(n^{2/3})$, then $ex(n, m, \{C_4, C_6 \}) = O(n)$. This conjecture was disproven by several papers and the current best-known bounds for this problem are $$ c_1n^{1 + \frac{1}{15}} \leq ex(n,…

In this article, we give a new upper bound for the regularity of edge ideals of gap-free graphs, in terms of the their minimal triangulation. Let $H_U=G\cup F_U$ be a minimal triangulation of a gap-free graph $G$, for some maximal…

We present a new algorithm for improving lower bounds on $ex(n;\{C_3,C_4\})$, the maximum size (number of edges) of an $n$-vertex graph of girth at least 5. The core of our algorithm is a variant of a hill-climbing heuristic introduced by…

The $3$-path isolation number of a connected $n$-vertex graph $G$, denoted by $\iota(G,P_3)$, is the size of a smallest subset $D$ of the vertex set of $G$ such that the closed neighbourhood $N[D]$ of $D$ in $G$ intersects each $3$-vertex…