Fort and Hedlund determined the minimum size of a $(v,3,2)$-covering design: [Minimal coverings of pairs by triples, Pacific J. Math. 8(4), 709-719, 1958.][1] 

The case $(v,4,2)$ was solved by Mills: On the covering of pairs by quadruples [I, JCTA 13, 55–78, 1972][2] and [II, JCTA 15, 138–166 (1973)][3].

Caro and Yuster proved that for sufficiently large $v$ the minimum size of a $(v,k,2)$ covering design is $\left\lceil\frac{v}{k}\left\lceil \frac{v-1}{k-1}\right\rceil\right\rceil$: [Covering Graphs: The Covering Problem Solved, JCTA 83 (2) 273–282, 1998][4].

The [La Jolla Covering repository][5] is a good source for covering designs. According to [this table][6] the smallest open case with $t=2$ is $(v,k,t)=(23,6,2)$ where the minimum size is either 20 or 21.
 


  [1]: http://projecteuclid.org/euclid.pjm/1103039697
  [2]: http://www.sciencedirect.com/science/article/pii/0097316572900088
  [3]: http://www.sciencedirect.com/science/article/pii/S0097316573800032
  [4]: http://www.sciencedirect.com/science/article/pii/S0097316598928679
  [5]: https://www.ccrwest.org/cover.html
  [6]: http://www.ccrwest.org/cover/low_tab.html