Poly time superset of NP complete language with infinitely many strings excluded from it For any arbitrary NP complete language is there always a polytime superset the complement of which is also infinite?

A trivial version which does not stipulate the superset to have infinite complement has been asked at https://cs.stackexchange.com/q/50123/42961

For purposes of this question, you can assume that $P \ne NP$.  As Vor explained, if $P = NP$ then the answer is "No".  (If $P = NP$, then $X = \{x \mid x \in \mathbb{N^+} \land x &gt; 1\}$ is NP-complete. Clearly there is no superset of $X$ which is infinite and has an infinite complement, as the complement of $X$ has only a single element.)  Thus we can focus on the case $P \ne NP$.
Every $\mathsf{coNP}$-complete set contains an infinite subset in $\mathsf{P}$ assuming that 


pseudorandom generators exist, and 
secure one-way permutations exist.


In other words, assuming that these two conjectures are true, 
no $\mathsf{coNP}$-complete set is P-immune.
As pointed out in the comments by Lance, 
this is implied by Theorem 4.4 of 


Glasser, Pavan, Selman, and Sengupta, "Properties of NP-complete sets",
SIAM J. Comput. 36(2), 516–542.


(Kaveh has already shown that your question is equivalent to 
whether every $\mathsf{coNP}$-complete sets 
contains an infinite $\mathsf{P}$ subset. 
In other language, 
this is saying that no $\mathsf{coNP}$-complete set is "$\mathsf{P}$-immune." This is the language used in the above-referenced theorem.)