6 research outputs found
Set systems without a 3-simplex
A 3-simplex is a collection of four sets A_1,...,A_4 with empty intersection
such that any three of them have nonempty intersection. We show that the
maximum size of a set system on n elements without a 3-simplex is for all , with
equality only achieved by the family of sets either containing a given element
or of size at most 2. This extends a result of Keevash and Mubayi, who showed
the conclusion for n sufficiently large.Comment: 5 page
The diamond-free process
Let K_4^- denote the diamond graph, formed by removing an edge from the
complete graph K_4. We consider the following random graph process: starting
with n isolated vertices, add edges uniformly at random provided no such edge
creates a copy of K_4^-. We show that, with probability tending to 1 as , the final size of the graph produced is . Our analysis also suggests that the graph produced after i edges are
added resembles the random graph, with the additional condition that the edges
which do not lie on triangles form a random-looking subgraph.Comment: 25 page