Related papers: A note on Turan number T(k+1,k,n)
We obtain upper bounds on the number of finite sets $\mathcal S$ of primes below a given bound for which various $2$ variable $\mathcal S$-unit equations have a solution.
Given a fixed graph H, we say that a graph G is H-free if G does not contain H as a subgraph. The Tur\'an number ex(n, H) of H is the maximum number of edges in an n-vertex H-free graph. The study of Tur\'an number of graphs is a central…
In this paper we establish an explicit upper bound for the first $k$-Ramanujan prime $R_1^{(k)}$ by using a recent result concerning the existence of prime numbers in small intervals.
The regular Tur\'an number of a graph $F$, denoted by rex$(n,F)$, is the largest number of edges in a regular graph $G$ of order $n$ such that $G$ does not contain subgraphs isomorphic to $F$. Giving a partial answer to a recent problem…
In this paper we confirm a conjecture of F\"uredi, Jiang, and Seiver, and determine an exact formula for the Tur\'an number $ex_3(n; P_3^3)$ of the 3-uniform linear path $P^3_3$ of length 3, valid for all $n$. It coincides with the…
A graph is outerplanar if it has a planar drawing for which all vertices belong to the outer face of the drawing. Let $H$ be a graph. The outerplanar Tur\'an number of $H$, denoted by $ex_\mathcal{OP}(n,H)$, is the maximum number of edges…
Very recently, Alon and Frankl initiated the study of the maximum number of edges in $n$-vertex $F$-free graphs with matching number at most $s$. For fixed $F$ and $s$, we determine this number apart from a constant additive term. We also…
Given a graph H and a positive integer n, the planar Turan number of H, denoted by exp(n, H), is the maximum number of edges in an n-vertex H-free planar graph.D.Ghosh, et al.initiated the topic of double stars S_(k,l). Recently Xu et…
This paper is motivated by the question of how global and dense restriction sets in results from extremal combinatorics can be replaced by less global and sparser ones. The result we consider here as an example is Turan's theorem, which…
Let $\mathcal{H}$ be an $r$-uniform hypergraph and $F$ be a graph. We say $\mathcal{H}$ contains $F$ as a trace if there exists some set $S\subseteq V(\mathcal{H})$ such that $\mathcal{H}|_{S}:=\{E\cap S: E\in E(\mathcal{H})\}$ contains a…
Write $K^{(k)}_{n}$ for the complete $k$-graph on $n$ vertices. For $2 \leq k \leq g < r$ integers, let $\pi\left(n, K^{(k)}_{g}, K^{(k)}_r\right)$ be the maximum density of $K^{(k)}_{g}$ in $n$ vertex $K^{(k)}_{r}$-free $k$-graphs. The…
In the present paper, we study problems related to the classical Borsuk's problem. Recall that the Borsuk's problem consists in finding the smallest number $ f(n) $ of parts of smaller diameter into which an arbitrary set of diameter 1 in…
Let $\mathcal{F}$ be a family of $r$-graphs. The Tur\'an number $ex_r(n;\mathcal{F})$ is defined to be the maximum number of edges in an $r$-graph of order $n$ that is $\mathcal{F}$-free. The famous Erd\H{o}s Matching Conjecture shows that…
The concept of $t$-$(v,k)$ trades of block designs previously has been studied in detail. See for example A. S. Hedayat (1990) and Billington (2003). Also Latin trades have been studied in detail under various names, see A. D. Keedwell…
In this note we revisit the problem of determining combinatorially the multiplicity at the origin of a toric curve. In addition, we give the exact value of the regularity index of that point for plane toric curves and effective bounds for…
The combinatorial theory for the set of parity alternating permutations is expounded. In view of the numbers of ascents and inversions, several enumerative aspects of the set are investigated. In particular, it is shown that signed Eulerian…
We give an upper bound for the dealternating number of a closed 3-braid. As applications, we determine the dealternating numbers, the alternation numbers and the Turaev genera of some closed positive 3-braids. We also show that there exist…
For graphs $H$ and $F$, the generalized Tur\'an number $ex(n,H,F)$ is the largest number of copies of $H$ in an $F$-free graph on $n$ vertices. We say that $H$ is $F$-Tur\'an-good if $ex(n,H,F)$ is the number of copies in the…
In a generalized Tur\'an problem, we are given graphs $H$ and $F$ and seek to maximize the number of copies of $H$ in an $F$-free graph of order $n$. We consider generalized Tur\'an problems where the host graph is planar. In particular we…
Tangent numbers $T_{2n-1}$, which enumerate alternating permutations of odd length, play a prominent role in the Taylor series expansion of the tangent function $\tan(x)$. In this work, we adopt a combinatorial approach based on the…