Beyond the Erdős Matching Conjecture
share
A family F⊂[n] k is U(s,q) of for any F_1,..., F_s∈ F we have |F_1∪...∪ F_s|< q. This notion generalizes the property of a family to be t-intersecting and to have matching number smaller than s. In this paper, we find the maximum | F| for F that are U(s,q), provided n>C(s,q)k with moderate C(s,q). In particular, we generalize the result of the first author on the Erdős Matching Conjecture and prove a generalization of the Erdős-Ko-Rado theorem, which states that for n> s^2k the largest family F⊂[n] k with property U(s,s(k-1)+1) is the star and is in particular intersecting. (Conversely, it is easy to see that any intersecting family in [n] k is U(s,s(k-1)+1).) We investigate the case k=3 more thoroughly, showing that, unlike in the case of the Erdős Matching Conjecture, in general there may be 3 extremal families.
READ FULL TEXT