Related papers: New Upper Bound for the Edge Folkman Number Fe(3,5…

Let $F$ be a strictly balanced $r$-uniform hypergraph with $e>2$ edges and $r$-density $m$. We give a new short proof of the fact that the Tur\'an number $\ex(n, F)$ is greater than $c\, n^{r-1/m} (\log n)^{1/(e-1)}$ where $c$ depends only…

The planar Tur\'{a}n number of a graph $H$, denoted by $ex_{\mathcal{P}}(n,H)$, is the maximum number of edges in an $n$-vertex $H$-free planar graph. Recently, D. Ghosh, et al. initiated the topic of double stars and prove that…

Let $f(d)$ be the smallest value for which every bridgeless graph $G$ with diameter $d$ admits a strong orientation $\overrightarrow{G}$ such that the diameter of $\overrightarrow{G}$ is at most $f(d)$. Chv\'atal and Thomassen (1978)…

We obtain some new upper bounds on the maximum number $f(n)$ of edges in $n$-vertex graphs without containing cycles of length four. This leads to an asymptotically optimal bound on $f(n)$ for a broad range of integers $n$ as well as a…

In an orientation $O$ of the graph $G$, an arc $e$ is deletable if and only if $O-e$ is strongly connected. For a $3$-edge-connected graph $G$, the Frank number is the minimum $k$ for which $G$ admits $k$ strongly connected orientations…

For a fixed graph $F$, we would like to determine the maximum number of edges in a properly edge-colored graph on $n$ vertices which does not contain a rainbow copy of $F$, that is, a copy of $F$ all of whose edges receive a different…

For a graph $G$ and integers $a_i \geq 1$, we say that $G \xrightarrow[]{} (a_1, \ldots, a_k)^v$ if in any $k$-coloring of $G$'s vertices there exists a monochromatic $a_i$-clique for some color $i \in \{1,\ldots,k\}$. $G \xrightarrow[]{}…

For an undirected simple graph $G$, we write $G \rightarrow (H_1, H_2)^v$ if and only if for every red-blue coloring of its vertices there exists a red $H_1$ or a blue $H_2$. The generalized vertex Folkman number $F_v(H_1, H_2; H)$ is…

In this note, we demonstrate that there is no [21, 5, 14] code over F5.

The vertex Folkman number $F_v(s,t;k)$ is the smallest $n$ for which there exists a $K_k$-free graph on $n$ vertices whose vertices cannot be $2$-colored without producing a monochromatic copy of $K_s$ or $K_t$. We show $F_v(3,3;5)=8$. The…

We prove that the number of 1324-avoiding permutations of length n is less than (7+4\sqrt{3})^n.

A linear $3$-graph is a set of vertices along with a set of edges, which are three element subsets of the vertices, such that any two edges intersect in at most one vertex. The crown, $C$, is a specific $3$-graph consisting of three…

Refining an existing counting argument, we provide an improved upper bound for the number of 1324-avoiding permutations of a given length.

In this paper we present three different results dealing with the number of $(\leq k)$-facets of a set of points: 1. We give structural properties of sets in the plane that achieve the optimal lower bound $3\binom{k+2}{2}$ of $(\leq…

Let $G$ be a graph and $a_1, ..., a_s$ be positive integers. Then $G \overset{v}{\rightarrow} (a_1, ..., a_s)$ means that for every coloring of the vertices of $G$ in $s$ colors there exists $i \in \{1, ..., s\}$, such that there is a…

We give upper bounds on weighted perfect matchings in pfaffian graphs. These upper bounds are better than Bregman's upper bounds on the number of perfect matchings. We show that some of our upper bounds are sharp for 3 and 4-regular…

Let g(n) denote the minimum number of edges of a maximal nontraceable graph of order n. Dudek, Katona and Wojda (2003) showed that g(n)\geq\ceil{(3n-2)/2}-2 for n\geq 20 and g(n)\leq\ceil{(3n-2)/2} for n\geq 54 as well as for n\in…

For a graph $G$ and integers $a_i\ge 1$, the expression $G \rightarrow (a_1,\dots,a_r)^v$ means that for any $r$-coloring of the vertices of $G$ there exists a monochromatic $a_i$-clique in $G$ for some color $i \in \{1,\cdots,r\}$. The…

Let $\mathcal{F}$ be a set of graphs. The planar Tur\'an number, $ex_{\mathcal{P}}(n,\mathcal{F})$, is the maximum number of edges in an $n$-vertex planar graph which does not contain any member of $\mathcal{F}$ as a subgraph. In this…

Let $r(k)$ denote the maximum number of edges in a $k$-uniform intersecting family with covering number $k$. Erd\H{o}s and Lov\'asz proved that $ \lfloor k! (e-1) \rfloor \leq r(k) \leq k^k.$ Frankl, Ota, and Tokushige improved the lower…