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We look at several saturation problems in complete balanced blow-ups of graphs. We let $H[n]$ denote the blow-up of $H$ onto parts of size $n$ and refer to a copy of $H$ in $H[n]$ as partite if it has one vertex in each part of $H[n]$. We then ask how few edges a subgraph $G$ of $H[n]$ can have such that $G$ has no partite copy of $H$ but such that the addition of any new edge from $H[n]$ creates a partite $H$. When $H$ is a triangle this value was determined by Ferrara, Jacobson, Pfender, and Wenger. Our main result is to calculate this value for $H=K_4$ when $n$ is large. We also give exact results for paths and stars and show that for $2$-connected graphs the answer is linear in $n$ whilst for graphs which are not $2$-connected the answer is quadratic in $n$. We also investigate a similar problem where $G$ is permitted to contain partite copies of $H$ but we require that the addition of any new edge from $H[n]$ creates an extra partite copy of $H$. This problem turns out to be much simpler and we attain exact answers for all cliques and trees.
Let $mathcal{C}$ be a family of edge-colored graphs. A $t$-edge colored graph $G$ is $(mathcal{C}, t)$-saturated if $G$ does not contain any graph in $mathcal{C}$ but the addition of any edge in any color in $[t]$ creates a copy of some graph in $mat
In 1964, ErdH{o}s, Hajnal and Moon introduced a saturation version of Turans classical theorem in extremal graph theory. In particular, they determined the minimum number of edges in a $K_r$-free, $n$-vertex graph with the property that the addition
The ErdH{o}s-Simonovits stability theorem states that for all epsilon >0 there exists alpha >0 such that if G is a K_{r+1}-free graph on n vertices with e(G) > ex(n,K_{r+1}) - alpha n^2, then one can remove epsilon n^2 edges from G to obtain an r-par
We study F-saturation games, first introduced by Furedi, Reimer and Seress in 1991, and named as such by West. The main question is to determine the length of the game whilst avoiding various classes of graph, playing on a large complete graph. We sh
By using the Szemeredi Regularity Lemma, Alon and Sudakov recently extended the classical Andrasfai-Erd~os-Sos theorem to cover general graphs. We prove, without using the Regularity Lemma, that the following stronger statement is true. Given any (r-