Hyper-optimistic conjecture
Gil Kalai and Tim Gowers have proposed a “hyper-optimistic” conjecture. Given a set [math]\displaystyle{ A \subset {}[3]^n }[/math], define the weighted size [math]\displaystyle{ \mu(A) }[/math] of [math]\displaystyle{ A }[/math] by the formula
- [math]\displaystyle{ \mu(A) := \sum_{a+b+c=n} |A \cap \Gamma_{a,b,c}|/|\Gamma_{a,b,c}| }[/math]
thus each slice [math]\displaystyle{ \Gamma_{a,b,c} }[/math] has weighted size 1 (and we have been referring to \mu as “slices-equal measure” for this reason), and the whole cube [math]\displaystyle{ {}[3]^n }[/math] has weighted size equal to the [math]\displaystyle{ (n+1)^{th} }[/math] triangular number, [math]\displaystyle{ \frac{(n+1)(n+2)}{2} }[/math].
Example: in [math]\displaystyle{ {}[3]^2 }[/math], the diagonal points 11, 22, 33 each have weighted size 1, whereas the other six off-diagonal points have weighted size 1/2. The total weighted size of [math]\displaystyle{ {}[3]^2 }[/math] is 6.
Let [math]\displaystyle{ c^\mu_n }[/math] be the largest weighted size of a line-free set. For instance, [math]\displaystyle{ c^\mu_0 = 1 }[/math], [math]\displaystyle{ c^\mu_1 = 2 }[/math], and [math]\displaystyle{ c^\mu_2 = 4 }[/math].
As in the unweighted case, every time we find a subset [math]\displaystyle{ B }[/math] of the grid [math]\displaystyle{ \Delta_n := \{ (a,b,c): a+b+c=n\} }[/math] without equilateral triangles, it gives a line-free set [math]\displaystyle{ \Gamma_B := \bigcup_{(a,b,c) \in B} \Gamma_{a,b,c} }[/math]. The weighted size of this set is precisely the cardinality of B. Thus we have the lower bound [math]\displaystyle{ c^\mu_n \geq \overline{c}^\mu_n }[/math], where [math]\displaystyle{ \overline{c}^\mu_n }[/math] is the largest size of equilateral triangles in [math]\displaystyle{ \Delta_n }[/math]. The computation of the [math]\displaystyle{ \overline{c}^\mu_n }[/math] is Fujimura's problem.
Hyper-optimistic conjecture: We in fact have [math]\displaystyle{ c^\mu_n = \overline{c}^\mu_n }[/math]. In other words, to get the optimal weighted size for a line-free set, one should take a set which is a union of slices [math]\displaystyle{ \Gamma_{a,b,c} }[/math].
This conjecture, if true, will imply the DHJ theorem. Note also that all our best lower bounds for the unweighted problem to date have been unions of slices. Also, the k=2 analogue of the conjecture is true, and is known as the LYM inequality (in fact, for k=2 we have [math]\displaystyle{ c^\mu_n = \overline{c}^\mu_n = 1 }[/math] for all n).
