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| :<math>\overline{c}^\mu_6 = 15</math>:
| | Let <math>\overline{c}^\mu_{n,4}</math> be the largest subset of the tetrahedral grid: |
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| <math>\overline{c}^\mu_6 \geq 15</math> from the bound for general n. | | :<math> \{ (a,b,c,d) \in {\Bbb Z}_+^4: a+b+c+d=n \}</math> |
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| Note that there are ten extremal solutions to <math>\overline{c}^\mu_3</math>:
| | which contains no tetrahedrons <math>(a+r,b,c,d), (a,b+r,c,d), (a,b,c+r,d), (a,b,c,d+r)</math> with <math>r > 0</math>; call such sets ''tetrahedron-free''. |
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| Solution I: remove 300, 020, 111, 003<br>
| | These are the currently known values of the sequence: |
| Solution II (and 2 rotations): remove 030, 111, 201, 102<br>
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| Solution III (and 2 rotations): remove 030, 021, 210, 102<br>
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| Solution III' (and 2 rotations): remove 030, 120, 012, 201<br>
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| Also consider the same triangular lattice with the point 020 removed, making a trapezoid. Solutions based on I-III are:
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| | | n || 0 || 1 || 2 |
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| | | <math>\overline{c}^\mu_{n,4}</math> || 1 || 3 || 7 |
| | |} |
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| Solution IV: remove 300, 111, 003<br>
| | == n=0 == |
| Solution V: remove 201, 111, 102<br>
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| Solution VI: remove 210, 021, 102<br>
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| Solution VI': remove 120, 012, 201<br>
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| The on the 7x7x7 triangular lattice triangle 141-411-114 must have at least one point removed. Remove 141, noting by symmetry any logic that follows will also work for either of the other two points.
| | <math>\overline{c}^\mu_{0,4} = 1</math>: |
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| Suppose we can remove all equilateral triangles on our 7×7x7 triangular lattice with only 12 removals.
| | There are no tetrahedrons, so no removals are needed. |
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| Here, "top triangle" means the top four rows of the lattice (with 060 at top) and "bottom trapezoid" means the bottom three rows.
| | == n=1 == |
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| At least 4 of those removals must come from the top triangle (the solutions of <math> \overline{c}^\mu_3</math> mentioned above).
| | <math>\overline{c}^\mu_{1,4} = 3</math>: |
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| The bottom trapezoid includes the overlapping trapezoids 600-420-321-303 and 303-123-024-006. If the solutions of these trapezoids come from V, VI, or VI', then 6 points have been removed. Suppose the trapezoid 600-420-321-303 uses the solution IV (by symmetry the same logic will work with the other trapezoid). Then there are 3 disjoint triangles 402-222-204, 213-123-114, and 105-015-006. Then 6 points have been removed. Therefore at least six removals must come from the bottom trapezoid.
| | Removing any one point on the grid will leave the set tetrahedron-free. |
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| To make a total of 12 removals there must be either:<br>
| | == n=2 == |
| Case A: 4 removals from the top triangle and 8 from the bottom trapezoid.<br>
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| Case B: 5 removals from the top triangle and 7 from the bottom trapezoid.<br>
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| Case C: 6 removals from the top triangle and 6 from the bottom trapezoid.<br>
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| * Suppose case A is true.
| | <math>\overline{c}^\mu_{2,4} = 7</math>: |
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| Because 141 is already removed, the solution to the upper triangle must remove either solution I (remove 060, 330, 033), solution II (remove 060, 231, 132), solution IIb (remove 033, 150, 240) or solution IIc (remove 330, 051, 042)
| | Suppose the set can be tetrahedron-free in two removals. One of (2,0,0,0), (0,2,0,0), (0,0,2,0), and (0,0,0,2) must be removed. Removing any one of the four leaves three tetrahedrons to remove. However, no point coincides with all three tetrahedrons, therefore there must be more than two removals. |
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| Suppose I is the solution for the upper triangle.
| | Three removals (for example (0,0,0,2), (1,1,0,0) and (0,0,2,0)) leaves the set tetrahedron-free with a set size of 7. |
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| Suppose 222 is open. Then 420, 321, 123, and 024 must all be removed. This leaves five disjoint triangles which require removals in the bottom trapezoid (150-600-105, 051-501-006, 222-402-204, 231-411-213, 132-312-114); therefore the bottom trapezoid needs at least 9 removals, but we can only make 8, therefore 222 is closed.
| | == General n == |
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| Suppose 411 is open. Then 213 and 015 must be removed. This leaves five disjoint triangles such that each triangle must have exactly one removal (420-150-123, 321-051,024, 600-510-501, 402-312-303, 204-114-105), so the remaining point (006) must be open, forcing 501 to be removed. This makes 600 and 510 open, and based on the triangles 600-240-204 and 510-150-114 both 204 and 114 must both be removed, but 204 and 114 are on the same disjoint triangle, contradicting the statement that each triangle must have exactly one removal. So 411 is closed.
| | A lower bound of 2(n-1)(n-2) can be obtained by keeping all points with exactly one coordinate equal to zero. |
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| This leaves six disjoint triangles each which must have at least one removal (420-123-150, 321-024-051, 510-213-240, 312-015-042, 501-204-231, 402-105-132). This forces 600 and 006 to be open. Based on the triangles 006-501-051 and 600-204-240, this forces 501 and 204 to be open. But then there are no removals from the triangle 501-204-231, which is a contradiction. Therefore the solution of the upper triangle cannot be I.
| | You get a non-constructive quadratic lower bound for the quadruple problem by taking a random subset of size <math>cn^2</math>. If c is not too large the linearity of expectation shows that the expected number of tetrahedrons in such a set is less than one, and so there must be a set of that size with no tetrahedrons. I think <math> c = \frac{24^{1/4}}{6} + o(\frac{1}{n})</math>. |
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| Suppose II is the solution for the upper triangle.
| | With coordinates (a,b,c,d), take the value a+2b+3c. This forms an arithmetic progression of length 4 for any of the tetrahedrons we are looking for. So we can take subsets of the form a+2b+3c=k, where k comes from a set with no such arithmetic progressions. [[http://arxiv.org/PS_cache/arxiv/pdf/0811/0811.3057v2.pdf This paper]] gives a complicated formula for the possible number of subsets. |
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| There are seven disjoint triangles (150-600-105, 051-501-006, 222-402-204, 240-510-213, 042-312-015, 330-420-321, 033-123-024), therefore of the three points remaining in the bottom trapezoid (411, 303, 114) exactly one must be removed.
| | One upper bound can be found by counting tetrahedrons. For a given n the tetrahedral grid has <math>\frac{1}{24}n(n+1)(n+2)(n+3)</math> tetrahedrons. Each point on the grid is part of n tetrahedrons, so <math>\frac{1}{24}(n+1)(n+2)(n+3)</math> points must be removed to remove all tetrahedrons. This gives an upper bound of <math>\frac{1}{8}(n+1)(n+2)(n+3)</math>. |
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| Suppose 411 is removed. Then 114 and 303 are open; 114 open forces 510 to be removed, forcing 213 to be open. 114 and 213 open force 123 to be closed, forcing 024 to be open. 024 open forces 222 to be closed, which forces 204 to be open, which leaves the triangle 213-303-204 open so we have a contradiction.
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| By symmetry 114 also can't be removed. Therefore we must remove 303. This leaves 411 and 114 open, forcing 510 and 015 closed. 510 and 015 closed forces 312 and 213 open, forcing 222 closed. 222 closed forces 402 and 204 open. 402 and 204 open force 501 and 105 closed. 510 and 105 closed force 600 and 006 open, leaving equilateral triangles 600-204-240 and 402-006-042 open. Therefore 303 can't be removed, and so the solution of the upper triangle can't be II.
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| Suppose IIb is the solution for the upper triangle.
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| Suppose 024 is open. This forces 420, 321, and 222 closed. Five disjoint triangles remain (510-600-501, 231-411-213, 132-312-114, 123-303-105, 024-204-006) so each must have exactly one point removed, and the remaining points in the lower trapezoid (402, 015) must be open. This forces 312, 411, 510, and 006 closed, which then force the the other points in their disjoint triangles open (600, 501, 231, 213, 132, 114, 204). The triangle 501-231-204 is therefore open, so we have a contradiction, therefore 024 is closed.
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| Suppose 006 is open. This forces 600, 501 and 402 to be closed, and leaves five disjoint triangles (510-420-411, 321-231-222, 312-132-114, 303-213-204, 105-015-006) and so 123 must be open. This forces 222 closed, which forces 321 open, which forces 420 closed, which forced 510 and 411 open, which forces 213 closed, which forces 303 and 204 open, which forces 105 closed, which forces 015 and 006 to be open, leaving an open triangle at 411-015-051. Therefore we have a contradiction, so 006 is closed.
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| Given 024 and 006 closed, now note there are six disjoint triangles (600-510-501, 402-312-303, 204-114-105, 420-330-321, 222-132-123, 411-231-213). Therefore the remaining point in the lower trapezoid 015 must be open, forcing 510, 411, and 312 to be closed. Using the disjoint triangles this forces 600, 501, 402, 303 and 213 to be open, which then forces 420 and 321 to be closed. Both 420 and 321 are on the same disjoint triangle, therefore we have a contradiction, so IIb can't be solution.
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| Note by symmetry, the same logic for IIb will apply for IIc. Therefore case A isn't true.
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| * Suppose case B is true.
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| The row 330-231-132-033 has ten possible solutions (excluding reflections, which by symmetry will be handled by the same logic):
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| Case Q: open-open-open-open<br>
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| Case R: closed-closed-closed-closed<br>
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| Case S: closed-open-open-open<br>
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| Case T: closed-open-closed-closed<br>
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| Case U: open-closed-closed-closed<br>
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| Case V: closed-open-closed-closed<br>
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| Case W: closed-closed-open-open<br>
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| Case X: closed-open-closed-open<br>
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| Case Y: open-closed-closed-open<br>
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| Case Z: closed-open-open-closed<br>
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| Consider all 10 cases. Note in all these cases we are still assuming 141 is removed.
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| (Case Q) 330, 231, 132, 033 open forces 060, 150, 051, 240, 141, 042 closed, but the upper triangle only allows 5 removals, so case Q forms a contradiction. | |
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| (Case R) 060-240-042 is left open, so case R forms a contradiction. | |
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| (Case S) 231, 132, and 033 open force 031 and 042 removed, which means from 240, 150, 061 there must be exactly one removal. | |
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| Suppose 213 is open. Then 411 and 015 are closed, and five disjoint triangles are left where each triangle must have exactly one removal (600-420-402, 501-321-303, 312-222-213, 204-105-114, 123-024-033). So the two remaining points in the lower trapezoid (510 and 006) must be open. 510 and 006 open force 303 closed, which forces 321 and 501 open, which forces 204 closed, which forces 114 and 103 open, which forces 402 and 312 closed, forcing 222 open, leaving 321-231-222 as an open triangle, so we have a contradiction.
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| Therefore 213 is closed. This leaves six disjoint triangles (600-420-402, 501-231-204, 303-033-006, 411-321-312, 222-132-123, 114-024-015) which each have exactly one removal. Therefore 510 from the lower trapezoid is open. Note since one of 150 and 060 must be open, then one of 114 or 015 must be closed; therefore 024 is open. This forces 123 closed, forcing 222 to be open, forcing 321 to be closed, forcing 411 and 312 open, forcing 421 closed, forcing 600 and 402 open, forcing 303 closed, forcing 006 open, forcing 204 closed, forcing 501 open, leaving the triangle 600-510-501 open and a contradiction.
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| (Case T) 231 is closed, and 330, 132, and 033 open force 060, 150, and 042 closed. Since 141 is already closed we have five removals from the top triangle, so 240 and 051 are open.
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| Suppose 024 is open. This forces 321 and 123 closed, leaving five disjoint triangles (600-510-501, 402-312-303, 204-114-105, 420-240-222, 213-033-015). Therefore 006 and 411 remaining in the lower trapezoid are open, forcing 501, 303 and 015 closed, forcing 600, 510, 312, 402, and 213 open, forcing 222 closed, forcing 420 open, leaving the open triangle 420-510-411, so we have a contradiction.
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| Therefore 024 is closed. There are now six disjoint triangles (510-600-501, 330-420-321, 312-402-303, 132-222-123, 033-213-015, 114-204-105). This leaves 411 and 006 open, which forces 015 and 501 closed, which forces 510 and 213 open, leaving the triangle 240-510-213 open, so we have a contradiction.
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| (Case U) Given the removals 231, 132, 033, and 141, the only possible removal to not leave any equilateral triangles in the upper triangle is 060. So 330, 240, 150, 051, and 042 are open.
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| Suppose 420 is open. This forces 321, 222, and 123 closed. This leaves five disjoint triangles (600-330-303, 510-420-411, 204-105-114, 501-051-006, and 312-042-015), so we have a contradiction.
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| Therefore 420 is closed. This leaves six disjoint triangles (600-330-303, 501-051-006, 204-114-105, 510-240-213, 411-321-312, 222-042-024). 015 in the lower trapezoid is therefore open, forcing 312 and 411 closed, but 312 and 411 are on the same disjoint triangle, so we have a contradiction.
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| (Case V) Given the removals 330, 132, and 033, the only possible removal to not leave any equilateral triangles in the upper triangle is 060. So 150, 051, 240, 042, and 231 are open.
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| Suppose 420 is open. Then 222 and 123 are closed, and we are left with 6 disjoint triangles (150-600-105, 240-510-213, 231-501-204, 024-114-015, 042-402-006, 411-321-312). This exceeds our limit of 7 removals in the bottom trapezoid, so we have a contradiction.
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| Therefore 420 is closed. Again we are left with the same 6 disjoint triangles (150-600-105, 240-510-213, 231-501-204, 024-114-015, 042-402-006, 411-321-312). Therefore 222, 123, and 303 are open. This forces 321, 024, and 105 to be closed, which then forces 411 and 015 to be open, forming an open triangle at 051-411-015, so we have a contradiction.
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| (Case W) Given the removals 330 and 231 with 132 and 033 open, 132 and 033 open force 042 closed. Given 141 is already closed, that leaves one more removal on the upper triangle, which must be one of 060-150-051, so 240 must be open. | |
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| Suppose 024 is open. This forces 123 to be closed, and leaves 6 disjoint triangles (600-510-501, 420-240-222, 411-321-312, 402-132-105, 213-033-015, 204-024-006). So 303 and 114 in the lower trapezoid are left open, forcing 312 and 005 to be closed, forcing 204 and 321 to be open, forcing 600 and 501 to be closed. 600 and 501 are on the same disjoint triangle so we have a contradiction.
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| Therefore 024 is closed. Suppose 312 is open. This forces 114 to be closed, and leaevs 5 disjoint triangles (600-510-501, 411-321-312, 303-213-204, 222-132-123, 105-015-006). Therefore 402 in the lower trapezoid is open, which forces 105 to be closed, which forces 015 and 006 to be open, which forces 213 and 303 to be closed, which are on the same disjoint triangle, so we have a contradiction.
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| Therefore 312 is closed. This leaves five disjoint triangles (510-240-213, 501-411-402, 222-132-123, 204-114-105, 303-033-006), and leaves 600, 420, 321, and 015 in the lower trapezoid open. This forces 402 and 213 to be closed, which forces 510 and 411 to be open, leaving an open triangle at 510-420-411, so we have a contradiction.
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| (Case X) 330 and 132 closed and 231 and 033 open imply 051 is closed, and since 141 is closed and we have only one more removal in the upper triangle it must be one of 240, 042, or 060. This leaves 150 open.
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| Suppose 321 is open. This forces 222 to be closed and leaves six disjoint triangles (600-420-402, 510-150-114, 411-321-312, 303-213-204, 105-015-006, 123-033-024), leaving 501 in the lower trapezoid open. This forces 204 to be closed, which forces 303 to be open, leaving an open triangle at 501-321-303.
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| So 321 is closed. This leaves six disjoint triangles (600-420-402, 510-150-114, 501-231-204, 312-222-213, 123-033-024, 105-015-006) and leaves 303 and 411 in the lower trapezoid open. This forces 213 and 006 to be closed, and forces 312, 222, 105, and 015 to be open. This forces 600 to be closed and 402 to be open, leaving an open triangle at 402-312-303. So we have a contradiction.
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| (Case Y) 330 and 033 are open, 330 and 033 open force 060 to be closed. With 141 closed there can be only one more removal, so suppose 150 is open (and note that if 150 was closed, we can use symmetry considering 051 to be open).
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| Suppose 600 is open. This forces 303 to be closed, and leaves six disjoint triangles (510-150-114, 420-330-321, 411-501-402, 222-312-213, 033-123-024, 015-105-006). So 204 in the lower trapezoid is left open. 600 and 150 open implies 105 is closed, forcing 015 and 006 to be open, forcing 024 and 213 to be closed, forcing 312, 222, and 123 to be open, forcing 042 in the upper triangle to be closed (so 051 and 240 are open). 051 and 015 open force 411 to be closed, which forces 501 to be open and leaves the open triangle 051-501-006.
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| So 600 is closed. This leaves six disjoint triangles (510-150-114, 420-330-321, 411-501-402, 222-312-213, 033-123-024, 015-105-006). 015 is the lower trapezoid is then left open, forcing 312 and 411 to be closed. However, 312 and 411 are on the same disjoint triangle, so we have a contradiction.
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| (Case Z) 330, 141, and 033 are closed, and 231 and 132 are open. Suppose 051 is open (note that if this forms a contradiction, by symmetrical argument we can say both 150 and 051 are closed).
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| Suppose 114 is closed. This leaves six disjoint triangles (600-510-501, 402-312-303, 105-015-006, 411-231-213, 222-132-123, 321-051-024) and so 420 and 204 are open. This forces 501 to be closed, which forces 510 and 600 to be open, which forces 411 and 402 to be closed, which forces 213 and 303 to be open, leaving an open triangle at 213-303-204.
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| So 114 is open. Suppose 213 is closed. This leaves five disjoint triangles (600-420-402, 501-321-303, 411-051-015, 222-132-123, 204-024-006) and in the lower trapezoid 510 and 105 are open. This forces 204 to be closed, forcing 024 and 006 to be open, forcing 015 to be closed, forcing 411 to be open, forcing 420 to be closed, forcing 402 to be open, leaving the open triangle 402-132-105.
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| So both 114 and 213 are open. Therefore 411 and 312 are closed. This leaves five disjoint triangles (600-510-501, 303-213-204, 105-015-006, 222-132-123, 321-051-024). The remaining points in the lower trapezoid (420, 402) are then open, forcing 105 to be closed, forcing 015 and 006 to be open, forcing 024 to be closed. Also note 114 and 213 open force 123 to be closed, which forces 222 to be open, which forces 321 to be closed. 321 and 024 are on the same disjoint triangle, so we have a contradiction.
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| So 150 and 051 are both closed. However, this leaves the triangle 240-042-060 open, so case Z is impossible.
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| Since all ten solutions have been eliminated, case B is impossible.
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| * Suppose case C is true.
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| Suppose the trapezoid 600-420-321-303 used solution IV. There are three disjoint triangles 402-222-204, 213-123-114, and 105-015-006. The remainder of the points in the lower trapezoid (420, 321, 510, 501, 402, 312, 024) must be left open. 024 being open forces either 114 or 015 to be removed.
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| Suppose 114 is removed. Then 213 is open, and with 312 open that forces 222 to be removed. Then 204 is open, and with 024 that forces 006 to be removed. So the bottom trapezoid is a removal configuration of 600-411-303-222-114-006, and the rest of the points in the bottom trapezoid are open. All 10 points in the upper triangle form equilateral triangles with bottom trapezoid points, hence 10 removals in the upper triangle would be needed (more than the 6 allowed), so 114 being removed doesn't work.
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| Suppose 015 is removed. Then 006-024 forces 204 to be removed. Regardless of where the removal in 123-213-114, the points 420, 321, 222, 024, 510, 312, 501, 402, 105, and 006 must be open. This forces upper triangle removals at 330, 231, 042, 060, 051, 132, our remaining 6 removals on the top triangle. However, we have already removed 141, forcing one removal too many, so the trapezoid 600-420-321-303 doesn't use solution IV.
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| Suppose the trapezoid 600-420-321-303 uses solution VI. The trapezoid 303-123-024-006 can't be IV (already eliminated by symmetry) or VI' (leaves the triangle 402-222-204). Suppose the trapezoid 303-123-024-006 is solution VI. The removals from the lower trapezoid are then 420, 501, 312, 123, 204, and 015, leaving the remaining points in the lower trapezoid open. The remaining open points is forces 10 upper triangle removals, so the trapezoid 600-420-321-303 doesn't use solution VI. Therefore the trapezoid 303-123-024-006 is solution V. The removals from the lower trapezoid are then 420, 510, 312, 204, 114, and 105. The remaining points in the lower trapezoid are open, and force 9 upper triangle removals, hence the trapezoid 303-123-024-006 can't be V, and the solution for 600-420-321-303 can't be VI.
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| The solution VI' for the trapezoid 600-420-321-303 can be eliminated by the same logic by symmetry.
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| Therefore it is impossible for the bottom trapezoid to use only 6 removals.
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| We have determined cases A, B, and C to be impossible, therefore it impossible to form a triangle free configuration on the 7x7x7 lattice with only 12 removals. Therefore <math>\overline{c}^\mu_6 = 15</math>.
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Let [math]\displaystyle{ \overline{c}^\mu_{n,4} }[/math] be the largest subset of the tetrahedral grid:
- [math]\displaystyle{ \{ (a,b,c,d) \in {\Bbb Z}_+^4: a+b+c+d=n \} }[/math]
which contains no tetrahedrons [math]\displaystyle{ (a+r,b,c,d), (a,b+r,c,d), (a,b,c+r,d), (a,b,c,d+r) }[/math] with [math]\displaystyle{ r \gt 0 }[/math]; call such sets tetrahedron-free.
These are the currently known values of the sequence:
| n |
0 |
1 |
2
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| [math]\displaystyle{ \overline{c}^\mu_{n,4} }[/math] |
1 |
3 |
7
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n=0
[math]\displaystyle{ \overline{c}^\mu_{0,4} = 1 }[/math]:
There are no tetrahedrons, so no removals are needed.
n=1
[math]\displaystyle{ \overline{c}^\mu_{1,4} = 3 }[/math]:
Removing any one point on the grid will leave the set tetrahedron-free.
n=2
[math]\displaystyle{ \overline{c}^\mu_{2,4} = 7 }[/math]:
Suppose the set can be tetrahedron-free in two removals. One of (2,0,0,0), (0,2,0,0), (0,0,2,0), and (0,0,0,2) must be removed. Removing any one of the four leaves three tetrahedrons to remove. However, no point coincides with all three tetrahedrons, therefore there must be more than two removals.
Three removals (for example (0,0,0,2), (1,1,0,0) and (0,0,2,0)) leaves the set tetrahedron-free with a set size of 7.
General n
A lower bound of 2(n-1)(n-2) can be obtained by keeping all points with exactly one coordinate equal to zero.
You get a non-constructive quadratic lower bound for the quadruple problem by taking a random subset of size [math]\displaystyle{ cn^2 }[/math]. If c is not too large the linearity of expectation shows that the expected number of tetrahedrons in such a set is less than one, and so there must be a set of that size with no tetrahedrons. I think [math]\displaystyle{ c = \frac{24^{1/4}}{6} + o(\frac{1}{n}) }[/math].
With coordinates (a,b,c,d), take the value a+2b+3c. This forms an arithmetic progression of length 4 for any of the tetrahedrons we are looking for. So we can take subsets of the form a+2b+3c=k, where k comes from a set with no such arithmetic progressions. [This paper] gives a complicated formula for the possible number of subsets.
One upper bound can be found by counting tetrahedrons. For a given n the tetrahedral grid has [math]\displaystyle{ \frac{1}{24}n(n+1)(n+2)(n+3) }[/math] tetrahedrons. Each point on the grid is part of n tetrahedrons, so [math]\displaystyle{ \frac{1}{24}(n+1)(n+2)(n+3) }[/math] points must be removed to remove all tetrahedrons. This gives an upper bound of [math]\displaystyle{ \frac{1}{8}(n+1)(n+2)(n+3) }[/math].
