 
Summary: TurŽan's theorem in the hypercube
Noga Alon
Anja Krech
Tibor SzabŽo
February 18, 2007
Abstract
We are motivated by the analogue of TurŽan's theorem in the hypercube Qn: how
many edges can a Qdfree subgraph of Qn have? We study this question through its
Ramseytype variant and obtain asymptotic results. We show that for every odd d it
is possible to color the edges of Qn with (d+1)2
4 colors, such that each subcube Qd is
polychromatic, that is, contains an edge of each color. The number of colors is tight up
to a constant factor, as it turns out that a similar coloring with d+1
2 + 1 colors is not
possible. The corresponding question for vertices is also considered. It is not possible to
color the vertices of Qn with d + 2 colors, such that any Qd is polychromatic, but there
is a simple d + 1 coloring with this property. A relationship to antiRamsey colorings is
also discussed.
We discover much less about the TurŽantype question which motivated our investi
gations. Numerous problems and conjectures are raised.
