# Hyper-optimistic conjecture

Gil Kalai and Tim Gowers have proposed a “hyper-optimistic” conjecture.

Let $c^\mu_n$ be the equal-slices measure 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 equal-slices measure 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 equal-slices measure 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).

## Small values of $c^\mu_n$

I have now found the extremal solutions for the weighted problem in the hyper-optimistic conjecture, again using integer programming.

The first few values are

• $c^\mu_0=1$ (trivial)
• $c^\mu_1=2$ (trivial)
• $c^{\mu}_2=4$ with 3 solutions
• $c^{\mu}_3=6$ with 9 solutions
• $c^{\mu}_4=9$ with 1 solution
• $c^{\mu}_5=12$ with 1 solution

Comparing this with the known bounds for $\overline{c}^\mu_n$ we see that the hyper-optimistic conjecture is true for $n \leq 5$.