On extensions of some block designs

详细

There are some results concerning t-designs in which the number of points in the intersection of two blocks takes less than t values. For example, if t = 2, then the design is symmetric (in such a design, v = b or, equivalently, k = r). In 1974, B. Gross described t-(v, k, l) designs that, for some integer s, 0 < s < t, do not contain two blocks intersecting at exactly s points. Below, it is proved that potentially infinite series of designs from the claim of Gross’ theorem are finite. Gross’ theorem is substantially sharpened.