Consider the set $S=\{1,2,\ldots, n\}$, and let $a<b<n$. What is the minimum number $f(a,b)$ such that there exist $f(a,b)$ subsets of $S$ of size $a$ for which any subset of $S$ of size $b$ contains at least one of the chosen subsets?


It is not hard to obtain a bound $f(a,b)\leq \dbinom{n}{a}-\dbinom{b}{a}+1$, as is shown [here](https://math.stackexchange.com/questions/967253/choosing-subsets-to-cover-larger-sets). But it doesn't seem easy to go beyond that. However, since this is such a simply-stated question, I would expect that it has been well-studied in combinatorial set theory. So, any references to better upper or lower bounds (or better yet, the exact answer) would be appreciated.



Choosing subsets to cover larger sets
This is Turán's problem. I quote the opening paragraph of A. E. Brouwer and M. Voorhoeve, "Turán theory and the lotto problem", Mathematical Center Tracts 106 (1979), 99-105 = Chapter 7 of A. Schrijver, ed., *Packing and Covering in Combinatorics*, Mathematisch Centrum, Amsterdam, 1979, ISBN 90-6196-180-7.



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> Let $k,\ell,n\in\mathbb N$ such that $k\le\ell\le n$. We define the *Turán number* $T(n,k,\ell)$ as the smallest number of $k$-subsets of an $n$-set $X$ such that any $\ell$-subset of $X$ contains at least one of these $k$-subsets. For example: $T(7,4,5)=7$. (Take $X=\{0,1,\dots,6\}$; the $4$-subsets are all translates $\pmod7$ of $\{1,2,3,5\}$; this is easily seen to be optimal.) The relation between Turán numbers and covering numbers is discussed in Chapters 4 and 5. The above definition can be formulated in the language of hypergraphs (see Chapter 1) as follows: for a hypergraph $H=(X,\mathcal E)$, let its *stability* number $\beta(H)$ be the maximal cardinality of a stable subset of $H$ (i.e. a set containing no edge). Then $T(n,k,\ell)$ is the minimal number of edges of a $k$-uniform hypergraph $H$ with $n$ vertices such that $\beta(H)\lt\ell$.  
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>  P. TURÁN [10] posed the problem of determining $T(n,k,\ell)$. In this section we give some estimates for this number. Notice that $T(n,k,\ell)$ is increasing in $n$ and $k$ and decreasing in $\ell$. Trivially, $T(n,1,\ell)=n-\ell+1$. The numbers $T(n,2,\ell)$ and the corresponding graphs are determined by the following theorem of TURÁN [9].
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