Time complexity of set intersection This problem involves the time-complexity of determining set intersections, and the  algorithm must give output on all possible inputs (as described below).

Problem 1:  The input  is a positive integer $m$, and two  unordered subsets $A$ and $B$ of $\{1,\dots,n\}$. The size of the input is $n+ |A| +|B|$.  

Output: The set $A \cap B$, the intersection of $A$ and $B$.

As all  infinitely many  possibilities  are ranged through,  with  the size of output $n = m + |A| + |B|$,   is there a linear-time  algorithm that on any $(n,A,B)$, outputs  $A \cap B$? Is there an $n P(\log n)$ algorithm, where $P$ is some polynomial with integer coefficients? (Worst-case complexity.) 

Problem 2:  The input  is a positive integer $m$, and  $j$ unordered subsets $A_1, \dots, A_j$  of $\{1,\dots,n\}$. The size of the input is $n+ |A_1| + \cdots + |A_n|$.  

Output: $A_1 \cap  A_2 \cap \cdots \cap A_n$, the intersection of $A_1, \dots, A_j$. 

As all infinitely many  possibilities  are ranged through,  with  the size of output $n = m + |A| + |B|$,   is there a linear-time  algorithm that on any $(n,A,B)$, outputs  $A \cap B$? Is there an $n P(\log n)$ algorithm, where $P$ is some polynomial with integer coefficients?  (Worst-case complexity.) 

Are there references to the time complexity of these problems? I’m interested in the particular algorithms themselves, but if anyone knows that complexities above are linear-time or $n P(\log n)$ algorithm, where $P$ is some polynomial with integer coefficients, I’d be grateful to hear from you.  ( I’m less interested in algorithms that involve hash functions, but as long as such an algorithm works on all possible inputs, a hash function algorithm is okay.)   I’m not a computer scientist student but an older person learning things as they come along in work-in-progress.
Your post contains two problems. I will only address the first.

There is a simple $O(n)$ algorithm for computing $A \cap B$, which involves an array of length $n$.

Another algorithm sorts $A \cup B$ to find the intersection in time $O(n\log n)$. In contrast to the previous algorithm, this algorithm can be implemented using only comparisons. There is a matching $\Omega(n\log n)$ lower bound in the algebraic decision tree model; see for example lecture notes of Otfried Cheong.