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97c3405 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 | import OAI.Combinatorics.CompleteCrossing.Counting
import OAI.Combinatorics.CompleteCrossing.CycleIdentity
import OAI.Combinatorics.CompleteCrossing.Deletion
import OAI.Combinatorics.CompleteCrossing.LowerBound
import OAI.Combinatorics.CompleteCrossing.Normalization
import OAI.Combinatorics.CompleteCrossing.TwoPage
namespace OAI
/-! Odd and even lower bounds, explicit attainment, and the ordinary complete-graph crossing formula. -/
section
noncomputable section
open scoped BigOperators Classical
namespace Paper170
theorem complete_graph_odd_lower_bound (s : ℕ) (hs : 2 ≤ s)
(N : NormalizedDrawing (completeGraph (2 * s + 1))) :
(s.choose 2) ^ 2 ≤ crossingCount N.toAdmissibleDrawing := by
obtain ⟨O⟩ := spoke_orders_exist _ N
obtain ⟨orders, hlocal⟩ := complete_local_orders N O
obtain ⟨hleft, hright, hrevleft, hrevright, hskew⟩ := raw_cycle_symmetries N O
let node : Fin (2 * s + 1) → ℝ := fun i => i.val
have hn : Function.Injective node := by
intro i j h
exact Fin.ext (Nat.cast_injective h)
have ht (H : Poly4) (hH : H ∈ box4 (2 * s - 1)) :
polynomialTest node (rawCycleIntersection O) H = 0 := by
apply polynomial_cycle_tests (by omega) node hn (rawCycleIntersection O)
(raw_cycle_pairing N O) H
convert hH using 1
congr 1
have hl := support_cardinality_lower_bound hs node hn (rawCycleIntersection O) orders
hleft hright hrevleft hrevright hskew hlocal ht
have hu := signed_support_bound N O
omega
lemma complete_graph_odd_lower_arbitrary (s : ℕ) (hs : 2 ≤ s)
(D : AdmissibleDrawing (completeGraph (2 * s + 1))) : hill (2 * s + 1) ≤ crossingCount D := by
obtain ⟨N, hN⟩ := polygonal_no_adjacent_normalization _ D
rw [hill_odd]
exact (complete_graph_odd_lower_bound s hs N).trans hN
lemma complete_graph_even_lower_normalized (s : ℕ) (hs : 2 ≤ s)
(N : NormalizedDrawing (completeGraph (2 * s + 2))) :
hill (2 * s + 2) ≤ crossingCount N.toAdmissibleDrawing := by
obtain ⟨A, _, hA⟩ := vertex_deletion_count (2 * s + 1) (by omega) N
have hl : ∀ v, hill (2 * s + 1) ≤ crossingCount (A v) :=
fun v => complete_graph_odd_lower_arbitrary s hs (A v)
have hh := Finset.sum_le_sum (s := Finset.univ) (fun v _ => hl v)
rw [hA] at hh
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, smul_eq_mul] at hh
have he : 2 * (s + 1) = 2 * s + 2 := by omega
have ho : 2 * s + 2 - 1 = 2 * s + 1 := by omega
conv_lhs => rw [← he]
apply even_lower_bound_of_deletion (s + 1) _ (by omega)
simpa only [he, ho, Nat.add_assoc] using hh
theorem complete_graph_lower_bound (n : ℕ) (hn : 3 ≤ n)
(D : AdmissibleDrawing (completeGraph n)) : hill n ≤ crossingCount D := by
by_cases hsmall : n < 5
· interval_cases n <;> norm_num [hill]
by_cases hodd : n % 2 = 1
· obtain ⟨s, rfl⟩ : ∃ s, n = 2 * s + 1 := ⟨n / 2, by omega⟩
exact complete_graph_odd_lower_arbitrary s (by omega) D
· obtain ⟨s, rfl⟩ : ∃ s, n = 2 * s + 2 := ⟨n / 2 - 1, by omega⟩
obtain ⟨N, hN⟩ := polygonal_no_adjacent_normalization _ D
exact (complete_graph_even_lower_normalized s (by omega) N).trans hN
theorem complete_graph_upper_bound (n : ℕ) (hn : 3 ≤ n) :
∃ D : AdmissibleDrawing (completeGraph n), crossingCount D = hill n := by
obtain ⟨D, hD⟩ := complete_graph_upper_drawing n
exact ⟨D, hD.trans (crossingTuples_card hn)⟩
theorem complete_graph_crossing_number (n : ℕ) (hn : 3 ≤ n) :
ordinaryCrossingNumber (completeGraph n) = hill n := by
obtain ⟨D, hD⟩ := complete_graph_upper_bound n hn
apply Nat.le_antisymm
· rw [← hD]
exact Paper170Geometry.ordinaryCrossingNumber_le_count D
· obtain ⟨A,hA,_⟩ := minimum_drawing_exists (completeGraph n)
rw [← hA]
exact complete_graph_lower_bound n hn A
end Paper170
end
end
end OAI
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