How can I find the smallest set of groups of $n$ elements such that every element is in the same group as every other at least once? Background: I'm working on a King of the Hill challenge for Programming Puzzles & Code Golf, and I've run into a problem with how I'm creating the individual matchups (groups of 4 entries).
Currently, I'm simply generating all combinations of 4 elements (in the combinatoric sense) of entries, but that gets really big really quickly:
$$x \choose 4$$

(from Wolfram Alpha)
Therefore, my question is: How can I choose subsets (of length $n$) of a set such that each element of that set appears in the same subset as every other element at least once, while maintaining a minimal amount of subsets?
I can figure this out for $n = 2$, since the solution is quite intuitive: simply combine the first element with every other element, combine the second element with every other element that comes after it, etc. For a set of length $L$, this yields:
$$\sum_{x=1}^{L-1}(L - x)$$

(from Wolfram Alpha)
which is much smaller and more manageable.
However, I still have not been able to figure out: How can I generalize this to any $n$?
 A: There is no simple formula for the general case.
What you are asking about are called block covering designs.  Specifically, a $(v,k,t)$-design is a collection of $k$-element subsets ("blocks") of (say) $\{1,2,\ldots,v\}$ such that any $t$-element subset is contained in at least one block.  Your "universe" has size $n$, but the convention in this area is to use $v$ for the total number of points/individuals.
Things are fairly well understood for $t=2$, which seems to be your "individual matchups".  A quick reference/database for known block covering designs is the (now migrated) La Jolla Covering Repository.
For example, with ten players (in groups of four), we can achieve all $45$ pairings using only nine blocks, much fewer than the easily computed but excessive $\binom{10}{4}=210$ collection of all possible foursomes.
A good starting point for constructing these designs for specific $(v,k,t)$ is Gordan et al New Constructions for Covering Designs (1995).  In many cases designs are known but not whether they achieve the minimum number of blocks, so there is an on-going search for optimal designs, especially for $t > 2$.
For the "dual" packing problems (no two players meet more than once) with groups of five and six players, @EdPegg has posted Social Golfer Problem - Quintets, asking what is known for certain universe sizes $v$.  No answers so far, but his Question does contain some informative links for smaller group sizes.
