The tripod packing problem has been extensively studied (see  for a short survey), but so far the gap between the best lower and upper bounds remain enormous. A simplified version of the problem consists of, rather than packing tripods in the integer cube $[n]^3$, packing them on the $r$-layered grid $[n]^2\times [r]$. In this talk, we essentially solve this problem for $r=1,2,3,4$.
 Boris Aronov, Vida Dujmović, Pat Morin, Aurélien Ooms and Luís Fernando Schultz Xavier da Silveira. ``More Turán-type Theorems for Triangles in Convex Point Sets''.