Abstract
We consider families of \(k\)-subsets of \(\{1, \dots, n\}\), where \(n\) is a multiple of \(k\), which have no perfect matching. An equivalent condition for a family \(\mathcal{F}\) to have no perfect matching is for there to be a blocking set, which is a set of \(b\) elements of \(\{1, \dots, n\}\) that cannot be covered by \(b\) disjoint sets in \(\mathcal{F}\). We are specifically interested in the largest possible size of a family \(\mathcal{F}\) with no perfect matching and no blocking set of size less than \(b\). Frankl resolved the case of families with no singleton blocking set (in other words, the \(b=2\) case) for sufficiently large \(n\) and conjectured an optimal construction for general \(b\). Though Frankl's construction fails to be optimal for \(k = 2, 3\), we show that the construction is optimal whenever \(k \ge 100\) and \(n\) is sufficiently large.Mathematics Subject Classifications: 05D05
Highlights
If F has no matching of size r, Sx,y(F ) has no matching of size r
This problem is closely related to the Erdos matching conjecture, which states the following
Kr−1 k is the number of k-subsets of [kr − 1], and n k n−r+1 k is the number of k-subsets of [n] which contain an element of [r − 1]
Summary
This result will be enough to resolve the k = 2 case (the case of an ordinary graph), where Theorem 1.6 does not apply but we can still find a different optimal family F. To do this we use techniques related to juntas, which were introduced in [1].
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
