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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.

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

For the case $(v,5,2)$ with $v\equiv 0\pmod 4$, Abel, Assaf, Bennett, Bluskov and Greig established that the Schönheim bound $\left\lceil\frac{v}{5}\left\lceil \frac{v-1}{4}\right\rceil\right\rceil$ is tight for $v\geqslant 28$ with 17 possible exceptions in the range $40\leqslant v\leqslant 280$: Pair covering designs with block size 5, Discrete Mathematics 307(14), 1776-1791, 2007.

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.

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

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.

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

For the case $(v,5,2)$ with $v\equiv 0\pmod 4$, Abel, Assaf, Bennett, Bluskov and Greig established that the Schönheim bound $\left\lceil\frac{v}{5}\left\lceil \frac{v-1}{4}\right\rceil\right\rceil$ is tight for $v\geqslant 28$ with 17 possible exceptions in the range $40\leqslant v\leqslant 280$: Pair covering designs with block size 5, Discrete Mathematics 307(14), 1776-1791, 2007.

The same authors have results for $(v,6,2)$: Pair covering and other designs with block size 6, J Comb Des 15(6), 511-533, 2007

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.

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

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.

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

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.

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

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.

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

For the case $(v,5,2)$ with $v\equiv 0\pmod 4$, Abel, Assaf, Bennett, Bluskov and Greig established that the Schönheim bound $\left\lceil\frac{v}{5}\left\lceil \frac{v-1}{4}\right\rceil\right\rceil$ is tight for $v\geqslant 28$ with 17 possible exceptions in the range $40\leqslant v\leqslant 280$: Pair covering designs with block size 5, Discrete Mathematics 307(14), 1776-1791, 2007.

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.

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

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.

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.

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

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.

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

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.

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