How to contruct a pseudorandom with a specific property How to construct a pseudorandom function $PRF$ with the following property:

Probability that $PRF(k_1,i)=PRF(k_2,i)$ is negligible, for all keys $k_1,k_2$ such that $k_1\neq k_2$

Does the PRF with the above property have to be Pseudorandom function tribe ensembles mentioned here or it can be any PRF with sufficiently large output range.
Depending on what you consider an acceptable approach, you could get there by having a random-looking function from your PRF domain to a prime-order cyclic group, and then use the permutation $x \mapsto x^k$ on the group.

Under reasonable assumptions and choice of group, this is a PRF. And since the key-dependent part is a permutation, two keys will never map the same $i$ to the same function value.

So this construction satisfies your requirements, but it is not a tribe ensemble.