The Turán number of a given graph $H$, denoted by $\mathrm{ex}(n,H)$, is the maximum number of edges in an $H$-free graph on $n$ vertices. Applying a well-known result of Hajnal and Szemerédi, we determine the Turán number $\text{ex}(n,K_p\cup K_q$) of a vertex-disjoint union of cliques $K_p$ and $K_q$ for all values of $n$.

In the first part of the paper we are concerned about finite sequences (over arbitrary symbols) $u$ for which $Ex(u,n)=O\left(n\right)$. The function $Ex(u,n)$ measures the maximum length of finite sequences over $n$ symbols which contain no subsequence of the type $u$. It follows from the result of Hart and Sharir that the containment $ababa\prec u$ is a (minimal) obstacle to $Ex(u,n)=O\left(n\right)$. We show by means of a construction due to Sharir and Wiernik that there is another obstacle to the linear growth. In the second part of the paper we investigate whether...

Given a graph H and an integer r ≥ 2, let G → (H,r) denote the Ramsey property of a graph G, that is, every r-coloring of the edges of G results in a monochromatic copy of H. Further, let $m\left(G\right)=ma{x}_{F\subseteq G}\left|E\left(F\right)\right|/\left|V\left(F\right)\right|$ and define the Ramsey density $m_{inf}(H,r)$ as the infimum of m(G) over all graphs G such that G → (H,r). In the first part of this paper we show that when H is a complete graph Kₖ on k vertices, then $m_{inf}(H,r)=(R-1)/2$, where R = R(k;r) is the classical Ramsey number. As a corollary we derive a new proof of the result credited to Chvatál...