Collision-resistant Pseudorandom function Assume we have a pseudorandom function: PRF(.)

We want the following property holds with a high probability: 

$\forall i, 0\leq i \leq n: PRF(k_1, i)\neq PRF(k_2,i)$

where $k_1$ and $k_2$ are two random keys picked independently. 



Question 1: What is the pseudorandom function with the above property called in the literature?

Question 2: Is there any paper using the above pseudorandom function?
The question seems that it could be implying that you actually intended to ask about probability of any given index not colliding, as opposed to the probability of every index simultaneously not colliding. If this is the case, then the answer is trivial: Two outputs of any PRF, even at the same index, are independent random variables by definition and therefore collide with probability $\frac{1}{n}$.

However, in the case that you really do mean the universal notion, the situation is just slightly more involved: Any PRF with an output size $a$ (in bits) and an input size $b$ has this property with probability approximately $1-2^{a-b}$.

To see this, let $\rho$ denote the probability that the desired property holds, and take the codomain to be of size $\alpha n$ where $\alpha$ is a constant.

$$\rho = \Pr(\forall i \in [0, n]: PRF(k_1, i)\neq PRF(k_2,i))$$
Assuming the values are i.i.d:
$$\rho = \Pr(PRF(k_1,i) \neq PRF(k_2,i))^{n+1}$$
Assuming the values are uniform:
$$\rho = \left(1-\frac{1}{\alpha n}\right)^{n+1}$$
Taking the limit as $n\to\infty$:
$$\rho \approx e^{-1/\alpha}$$
Which is within 5% of the true value when $n \geq 32$ and $\alpha \geq 1$. We assume $\rho = 1-\frac{1}{z}$ and solve for $\log_2\alpha$, which is by definition the number of output bits minus the number of input bits:
$$e^{-1/\alpha} = 1-\frac{1}{z}$$
$$\alpha = \left(\log\frac{z}{z-1}\right)^{-1}$$
$$\log_2\alpha = -\log_2\log\frac{z}{z-1}$$
Which is always within half of one bit to the true value, as long as $z \leq e^n$. We simplify again with another convenient approximation:
$$\log_2\alpha \approx \log_2 z$$
Which is at worst one half bit below the true value. Since the probability of your condition holding is determined by $z$, and $\alpha$ is the ratio between the cardinalities of the output and input domains, the probability of your condition holding is approximately proportional to the size of the output domain divided by the size of the input domain. $\square$