Difference between revisions of "Fujimura's problem"

From Polymath1Wiki
Jump to: navigation, search
(n=4)
Line 55: Line 55:
  
 
Let <math>S\subset \Delta_4</math> be a set without equilateral triangles. If <math>(0,0,4)\in S</math>, there can only be one of <math>(0,x,4-x)</math> and <math>(x,0,4-x)</math> in S for <math>x=1,2,3,4</math>. Thus there can only be 5 elements in S with <math>a=0</math> or <math>b=0</math>. The set of elements with <math>a,b>0</math> is isomorphic to <math>\Delta_2</math>, so S can at most have 4 elements in this set. So <math>|S|\leq 4+5=9</math>. Similar if S contain (0,4,0) or (4,0,0). So if <math>|S|>9</math> S doesn’t contain any of these. Also, S can’t contain all of <math>(0,1,3), (0,3,1), (2,1,1)</math>. Similar for <math>(3,0,1), (1,0,3),(1,2,1)</math> and <math>(1,3,0), (3,1,0), (1,1,2)</math>. So now we have found 6 elements not in S, but <math>|\Delta_4|=15</math>, so <math>S\leq 15-6=9</math>.
 
Let <math>S\subset \Delta_4</math> be a set without equilateral triangles. If <math>(0,0,4)\in S</math>, there can only be one of <math>(0,x,4-x)</math> and <math>(x,0,4-x)</math> in S for <math>x=1,2,3,4</math>. Thus there can only be 5 elements in S with <math>a=0</math> or <math>b=0</math>. The set of elements with <math>a,b>0</math> is isomorphic to <math>\Delta_2</math>, so S can at most have 4 elements in this set. So <math>|S|\leq 4+5=9</math>. Similar if S contain (0,4,0) or (4,0,0). So if <math>|S|>9</math> S doesn’t contain any of these. Also, S can’t contain all of <math>(0,1,3), (0,3,1), (2,1,1)</math>. Similar for <math>(3,0,1), (1,0,3),(1,2,1)</math> and <math>(1,3,0), (3,1,0), (1,1,2)</math>. So now we have found 6 elements not in S, but <math>|\Delta_4|=15</math>, so <math>S\leq 15-6=9</math>.
 +
 +
''Remark'': curiously, the best constructions for <math>c_4</math> uses only 7 points instead of 9.
  
 
== n=5 ==
 
== n=5 ==

Revision as of 20:43, 13 February 2009

Let [math]\overline{c}^\mu_n[/math] the largest subset of the triangular grid

[math]\Delta_n := \{ (a,b,c) \in {\Bbb Z}_+^3: a+b+c=n \}[/math]

which contains no equilateral triangles [math](a+r,b,c), (a,b+r,c), (a,b,c+r)[/math] with [math]r \gt 0[/math]; call such sets triangle-free. (It is an interesting variant to also allow negative r, thus allowing "upside-down" triangles, but this does not seem to be as closely connected to DHJ(3).) Fujimura's problem is to compute [math]\overline{c}^\mu_n[/math]. This quantity is relevant to a certain hyper-optimistic conjecture.

n 0 1 2 3 4 5
[math]\overline{c}^\mu_n[/math] 1 2 4 6 9 12

n=0

[math]\overline{c}^\mu_0 = 1[/math]:

This is clear.

n=1

[math]\overline{c}^\mu_1 = 2[/math]:

This is clear.

n=2

[math]\overline{c}^\mu_2 = 4[/math]:

This is clear (e.g. remove (0,2,0) and (1,0,1) from [math]\Delta_2[/math]).

n=3

[math]\overline{c}^\mu_3 \geq 6[/math]:

For the lower bound, delete (0,3,0), (0,2,1), (2,1,0), (1,0,2) from [math]\Delta_3[/math].

For the upper bound: observe that with only three removals each of these (non-overlapping) triangles must have one removal:

  • set A: (0,3,0) (0,2,1) (1,2,0)
  • set B: (0,1,2) (0,0,3) (1,0,2)
  • set C: (2,1,0) (2,0,1) (3,0,0)

Consider choices from set A:

  • (0,3,0) leaves triangle (0,2,1) (1,2,0) (1,1,1)
  • (0,2,1) forces a second removal at (2,1,0) [otherwise there is triangle at (1,2,0) (1,1,1) (2,1,0)] but then none of the choices for third removal work
  • (1,2,0) is symmetrical with (0,2,1)

n=4

[math]\overline{c}^\mu_4=9[/math]:

The set of all [math](a,b,c)[/math] in [math]\Delta_4[/math] with exactly one of a,b,c =0, has 9 elements and is triangle-free. (Note that it does contain the equilateral triangle (2,2,0),(2,0,2),(0,2,2), so would not qualify for the generalised version of Fujimura's problem in which [math]r[/math] is allowed to be negative.)

Let [math]S\subset \Delta_4[/math] be a set without equilateral triangles. If [math](0,0,4)\in S[/math], there can only be one of [math](0,x,4-x)[/math] and [math](x,0,4-x)[/math] in S for [math]x=1,2,3,4[/math]. Thus there can only be 5 elements in S with [math]a=0[/math] or [math]b=0[/math]. The set of elements with [math]a,b\gt0[/math] is isomorphic to [math]\Delta_2[/math], so S can at most have 4 elements in this set. So [math]|S|\leq 4+5=9[/math]. Similar if S contain (0,4,0) or (4,0,0). So if [math]|S|\gt9[/math] S doesn’t contain any of these. Also, S can’t contain all of [math](0,1,3), (0,3,1), (2,1,1)[/math]. Similar for [math](3,0,1), (1,0,3),(1,2,1)[/math] and [math](1,3,0), (3,1,0), (1,1,2)[/math]. So now we have found 6 elements not in S, but [math]|\Delta_4|=15[/math], so [math]S\leq 15-6=9[/math].

Remark: curiously, the best constructions for [math]c_4[/math] uses only 7 points instead of 9.

n=5

[math]\overline{c}^\mu_5=12[/math]:

The set of all (a,b,c) in [math]\Delta_5[/math] with exactly one of a,b,c=0 has 12 elements and doesn’t contain any equilateral triangles.

Let [math]S\subset \Delta_5[/math] be a set without equilateral triangles. If [math](0,0,5)\in S[/math], there can only be one of (0,x,5-x) and (x,0,5-x) in S for x=1,2,3,4,5. Thus there can only be 6 elements in S with a=0 or b=0. The set of element with a,b>0 is isomorphic to [math]\Delta_3[/math], so S can at most have 6 elements in this set. So [math]|S|\leq 6+6=12[/math]. Similar if S contain (0,5,0) or (5,0,0). So if |S| >12 S doesn’t contain any of these. S can only contain 2 point in each of the following equilateral triangles:

(3,1,1),(0,4,1),(0,1,4)

(4,1,0),(1,4,0),(1,1,3)

(4,0,1),(1,3,1),(1,0,4)

(1,2,2),(0,3,2),(0,2,3)

(3,2,0),(2,3,0),(2,2,1)

(3,0,2),(2,1,2),(2,0,3)

So now we have found 9 elements not in S, but [math]|\Delta_5|=21[/math], so [math]S\leq 21-9=12[/math].

General n

A lower bound for [math]\overline{c}^\mu_n[/math] is 2n for [math]n \geq 1[/math], by removing (n,0,0), the triangle (n-2,1,1) (0,n-1,1) (0,1,n-1), and all points on the edges of and inside the same triangle. In a similar spirit, we have the lower bound

[math]\overline{c}^\mu_{n+1} \geq \overline{c}^\mu_n + 2[/math]

for [math]n \geq 1[/math], because we can take an example for [math]\overline{c}^\mu_n[/math] (which cannot be all of [math]\Delta_n[/math]) and add two points on the bottom row, chosen so that the triangle they form has third vertex outside of the original example.

An asymptotically superior lower bound for [math]\overline{c}^\mu_n[/math] is 3(n-1), made of all points in [math]\Delta_n[/math] with exactly one coordinate equal to zero.

A trivial upper bound is

[math]\overline{c}^\mu_{n+1} \leq \overline{c}^\mu_n + n+2[/math]

since deleting the bottom row of a equilateral-triangle-free-set gives another equilateral-triangle-free-set. We also have the asymptotically superior bound

[math]\overline{c}^\mu_{n+2} \leq \overline{c}^\mu_n + \frac{3n+2}{2}[/math]

which comes from deleting two bottom rows of a triangle-free set and counting how many vertices are possible in those rows.

Another upper bound comes from counting the triangles. There are [math]\binom{n+2}{3}[/math] triangles, and each point belongs to n of them. So you must remove at least (n+2)(n+1)/6 points to remove all triangles, leaving (n+2)(n+1)/3 points as an upper bound for [math]\overline{c}^\mu_n[/math].

Asymptotics

The corners theorem tells us that [math]\overline{c}^\mu_n = o(n^2)[/math] as [math]n \to \infty[/math].

By looking at those triples (a,b,c) with a+2b inside a Behrend set, one can obtain the lower bound [math]\overline{c}^\mu_n \geq n^2 \exp(-O(\sqrt{\log n}))[/math].