Highly linked graphs
A graph with at least 2k vertices is said to be k-linked if, for any choice s1, . . . ,sk, t1, . . . ,tk. of 2k distinct vertices there are vertex disjoint paths P1, . . . ,Pk with Pi joining si to ti, 1≤i≤k. Recently Robertson and Seymour  showed that a graph G is k-linked provided its vertex connectivity k(G) exceeds 10k-√log2 k. We show here that k(G)≥22k will do.
Bollobás, B., & Thomason, A. (1996). Highly linked graphs. Combinatorica, 16 (3), 313-320. https://doi.org/10.1007/bf01261316