On the time complexity of finding independent sets in hypergraphs
The ubiquitous principle is that a generalized problem is at least as hard to the original problem. This rather trivial principle is applied as truism in most if not all papers.



Here is a specific version of that principle for the case of time-complexity.

Lemma. Let $A,B, C$ be three languages such that $A= B\cap C$ and $A\not=C$. ($C$ is considered as a generalization of $A$.) It takes polynomial time to determine whether $x\in B$. Then deciding $C$ is as least as hard as deciding $A$ modulo polynomial time. If $A$ is NP-complete and $C$ is in NP, then $C$ is NP-complete.

Proof. Let $\beta$ be a polynomial-time algorithm that decides $B$. Suppose we have an algorithm $\gamma$ that decides $C$. Let $\alpha$ be the following algorithm:


Given input string $x$, run $\beta$ to determine if $x\in B$. If not, return no. If yes, run $\gamma$ to determine if $x\in C$. If not, return no. Return yes otherwise.


It is routine to check that $\alpha$ return yes if and only if the input is in $A$, i.e., $\alpha$ decides $A$. Since $\beta$ takes polynomial time, deciding $C$ is at least as hard as deciding $A$ modulo polynomial time.

It follows immediately that if $A$ is NP-complete and $C$ is in NP, then $C$ is NP-complete.



Let 


$A=\{\langle G,s\rangle \mid \text{ there is an independent set of size }s\text{ in an ordinary graph G}\}$. (I use the phrase "ordinary graph" or simply "graph" to contrast with "hypergraph". "General graph" sounds more like hypergraph. Indeed, hypergraph is a generalization of an ordinary graph.)
Let $B=\{\langle G\rangle\mid G \text{ is an ordinary graph}\}$. 
Let $C=\{\langle G, s\rangle \mid \text{ there is an independent set of size }s\text{ in a hypergraph G}\}$.


The lemma tells us that deciding $C$ is at least as hard as deciding $A$.



Exercise 1. Explain that deciding whether a graph is 3-colorable is at least as hard as deciding whether a given planar-graph is 3-colorable. (In fact, both problems are NP-complete.)

Exercise 2. Prove that it is NP-complete to determine whether there is independent set of given size in a given triangle hypergraph. A triangle hypergraph is defined as a hypergraph where each edge is a set of 3 vertices.