Rainbow saturation and graph capacities

TL;DR: In this paper, the authors considered a variety of colored saturation problems and showed that the extremal graphs can be determined exactly and the order of magnitude for the color saturation function is O(n, \mathcal{C}_2(K_3)) for all edge-colored graphs.

TL;DR: In this article, the authors introduce the notion of rainbow saturation and the corresponding rainbow saturation number, which is the saturation version of the rainbow Tur\'an numbers whose systematic study was initiated by Keevash, Mubayi, Sudakov, and Verstra\"ete.

TL;DR: It is proved that $\operatorname{sat}_{t}\left(n, \mathfrak{R}{(K_r)}\right)=\Theta(n\log n)$, for any graph $H$ belonging to a large class of connected graphs and for any $t\geq e(H)$.

TL;DR: The upper bounds are improved and it is shown thatsat_t(n,\mathfrak{R}(P_l) is proved to be the minimum number of edges in a $t$-edge-colored graph on $n$ vertices which does not contain a rainbow copy of $F$, i.e., a copy of F all of whose edges receive a different color.

TL;DR: An asymptotic solution to several hard problems in extremal set theory within a unified, formally information-theoretic framework is given.

TL;DR: The problem of minimizing the number of edges in a maximal Ks-free subgraph of the Erdi¾?s-Renyi random graph was studied in this paper.

TL;DR: In stark contrast with the related problem for monochromatic subgraphs, wherein the saturation is always linear in n, it is proved that rainbow saturation numbers have a variety of different orders of growth.

TL;DR: Results on the size of weakly and strongly separating set systems and matrices, and on cross-intersecting systems, are proved and it is shown that the minimum number of edges in an oriented graph of order n with diameter 2 is (1+o(1)log"2n).