The tri­pod pack­ing prob­lem has been ex­ten­sively stud­ied (see [1] 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 $r=1,2,3,4$.

[1] 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''.