# Difference between revisions of "Hyper-optimistic conjecture"

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− | [http://gowers.wordpress.com/2009/02/08/dhj-quasirandomness-and-obstructions-to-uniformity/#comment-2114 Gil Kalai] and [http://gowers.wordpress.com/2009/02/08/dhj-quasirandomness-and-obstructions-to-uniformity/#comment-2119 Tim Gowers] have proposed a “hyper-optimistic” conjecture. | + | [http://gowers.wordpress.com/2009/02/08/dhj-quasirandomness-and-obstructions-to-uniformity/#comment-2114 Gil Kalai] and [http://gowers.wordpress.com/2009/02/08/dhj-quasirandomness-and-obstructions-to-uniformity/#comment-2119 Tim Gowers] have proposed a “hyper-optimistic” conjecture. |

− | + | Let <math>c^\mu_n</math> be the [[equal-slices measure]] of a line-free set. For instance, <math>c^\mu_0 = 1</math>, <math>c^\mu_1 = 2</math>, and <math>c^\mu_2 = 4</math>. | |

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

− | + | '''Hyper-optimistic conjecture:''' We in fact have <math>c^\mu_n = \overline{c}^\mu_n</math>. 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 <math>\Gamma_{a,b,c}</math>. | |

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− | '''Hyper-optimistic conjecture:''' We in fact have <math>c^\mu_n = \overline{c}^\mu_n</math>. In other words, to get the optimal | + | |

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 [http://en.wikipedia.org/wiki/Lubell-Yamamoto-Meshalkin_inequality LYM inequality] (in fact, for k=2 we have <math>c^\mu_n = \overline{c}^\mu_n = 1</math> for all n). | 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 [http://en.wikipedia.org/wiki/Lubell-Yamamoto-Meshalkin_inequality LYM inequality] (in fact, for k=2 we have <math>c^\mu_n = \overline{c}^\mu_n = 1</math> for all n). |

## Revision as of 17:33, 15 February 2009

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

Let [math]c^\mu_n[/math] be the equal-slices measure of a line-free set. For instance, [math]c^\mu_0 = 1[/math], [math]c^\mu_1 = 2[/math], and [math]c^\mu_2 = 4[/math].

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

**Hyper-optimistic conjecture:** We in fact have [math]c^\mu_n = \overline{c}^\mu_n[/math]. 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 [math]\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]c^\mu_n = \overline{c}^\mu_n = 1[/math] for all n).