# Hyper-optimistic conjecture

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Gil Kalai and Tim Gowers have proposed a “hyper-optimistic” conjecture. Given a set $A \subset {}^n$, define the weighted size $\mu(A)$ of $A$ by the formula

$\mu(A) := \sum_{a+b+c=n} |A \cap \Gamma_{a,b,c}|/|\Gamma_{a,b,c}|$

thus each slice $\Gamma_{a,b,c}\lt\math\gt has weighted size 1 (and we have been referring to \mu as “slices-equal measure” for this reason), and the whole cube \ltmath\gt{}^n\lt\math\gt has weighted size equal to the \ltmath\gt(n+1)^{th}\lt\math\gt triangular number, \ltmath\gt\frac{(n+1)(n+2)}{2}\lt\math\gt. '''Example:''' in \ltmath\gt{}^2$, 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 ${}^2$ is 6.

Let $c^\mu_n$ be the largest weighted size of a line-free set. For instance, $c^\mu_0 = 1$, $c^\mu_1 = 2$, and $c^\mu_2 = 4$.

As in the unweighted case, every time we find a subset $B$ of the grid $\Delta_n := \{ (a,b,c): a+b+c=n\}$ without equilateral triangles, it gives a line-free set $\Gamma_B := \bigcup_{(a,b,c) \in B} \Gamma_{a,b,c}$. The weighted size of this set is precisely the cardinality of B. Thus we have the lower bound $c^\mu_n \geq \overline{c}^\mu_n$, where $\overline{c}^\mu_n$ is the largest size of equilateral triangles in $\Delta_n$. The computation of the $\overline{c}^\mu_n$ is Fujimura's problem.

Hyper-optimistic conjecture: We in fact have $c^\mu_n = \overline{c}^\mu_n$. 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 $\Gamma_{a,b,c}$.

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 $c^\mu_n = \overline{c}^\mu_n = 1$ for all n).