Q1: The state of the art as reported in 2014, see arXiv:1406.2097, is that the only numbers which are known to be Tarski numbers of some groups are 4,5,6. Tarski numbers $<4$ are forbidden, which suggests that 7 is the answer to the question "What is the smallest non negative integer that we do not know yet is the Tarski number of a group?".

Q2: Recent progress, in addition to the references by Manuel Norman: In arXiv:1603.04212 it is shown that the Tarski number of Burnside groups is in the range $[6,14]$.

Q1: The state of the art as reported in 2014, see arXiv:1406.2097, is that the only numbers which are known to be Tarski numbers of some groups are 4,5,6. Tarski numbers $<4$ are forbidden, which suggests that 7 is the answer to the question "What is the smallest non negative integer that we do not know yet is the Tarski number of a group?".

Q2: Recent progress, in addition to the references by Manuel Norman: In arXiv:1603.04212 it is shown that the Tarski number of Burnside groups is in the range $[6,14]$.

Q1: The state of the art as reported in 2014, see arXiv:1406.2097, is that the only numbers which are known to be Tarski numbers of some groups are 4,5,6. Tarski numbers $<4$ are forbidden, which suggests that 7 is the answer to the question "What is the smallest non negative integer that we do not know yet is the Tarski number of a group?".

A study of the citation trail since 2014 gives no counterindication.

Q1: The state of the art as reported in 2014, see arXiv:1406.2097, is that the only numbers which are known to be Tarski numbers of some groups are 4,5,6. Tarski numbers $<4$ are forbidden, which suggests that 7 is the answer to the question "What is the smallest non negative integer that we do not know yet is the Tarski number of a group?".

Q2: Recent progress, in addition to the references by Manuel Norman: In arXiv:1603.04212 it is shown that the Tarski number of Burnside groups is in the range $[6,14]$.