Two questions, the first: What is the smallest non negative integer that we do not know yet is the Tarski number of a group?
The second question is the same as in the title: What is the latest progress on Tarski numbers?



Latest progress on Tarski numbers
**Q1:** The state of the art as reported in 2014, see [arXiv:1406.2097](https://arxiv.org/abs/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](https://arxiv.org/abs/1603.04212) it is shown that the Tarski number of Burnside groups is in the range $[6,14]$.


