IP-Szemerédi theorem

IP-Szemerédi theorem: If n is sufficiently large depending on [math]\displaystyle{ \delta \gt 0 }[/math], then any subset of [math]\displaystyle{ [2]^n \times [2]^n }[/math] of density at least [math]\displaystyle{ \delta }[/math] contains a corner [math]\displaystyle{ (x,y), (x+r,y), (x,y+r) }[/math], where x, y, x+r, y+r all lie in [math]\displaystyle{ [2]^n }[/math] and r lies in [math]\displaystyle{ [2]^n \backslash 0^n }[/math].

Implies the corners theorem, and hence Roth's theorem. Is implied in turn by the density Hales-Jewett theorem, and may thus be a simpler test case.

No combinatorial proof of this theorem is currently known.

Randall McCutcheon proposes a slightly weaker version of this theorem in which [math]\displaystyle{ [2]^n }[/math] is replaced by [math]\displaystyle{ [n]^n }[/math], but r is still constrained to [math]\displaystyle{ [2]^n \backslash 0^n }[/math].