On 5-Gons and 5-Holes

Abstract

Abstract.We consider an extension of a question of Erdős on the number of k-gons in a set of n points in the plane. Relaxing the convexity restriction we obtain results on 5-gons and 5-holes (empty 5-gons). In particular, we show a direct relation between the number of non-convex 5-gons and the rectilinear crossing number, provide an improved lower bound for the number of convex 5-holes any point set must contain, and prove that the number of general 5-holes is asymptotically maximized for point sets in convex position.

Lower bounds for the number of small convex k -holes

“…In the presentation of [11] the lower bound was improved to h 5 (n) ≥ . A slightly better bound h 5 (n) ≥ 3 n− 4 8 was presented in [3], which was then sharpened to h 5 (n) ≥ 3 7 (n − 11) in [4]. The latest and so far best bound of…”

“…Already in [4] a set of 12 points containing only three convex 5-holes has been presented. This implies h 5 (12) = 3 and therefore disproves Dehnhardt's conjecture of h 5 (12) = 4.…”

“…Hence, we have h 3|5 (n) = h 5 (n) and h 4|5 (n) = 2 · h 5 (n), for n ∈ {10, 11, 12}, which proves the lemma. (Recall that h 5 (10) = 1, h 5 (11) = 2, and h 5 (12) = 3, by [14,8,4]. )…”

“…In this section we further improve this bound to h 5 (n) ≥ 3 4 n − o(n). We start by fine-tuning the proof from [4], showing h 5 (n) ≥ 3 7 (n − 11) , by utilizing the results h 5 (10) = 1 [14], h 5 (11) = 2 [8], and h 5 (12) ≥ 3 [8]. Although this does not lead to an improved lower bound of h 5 (n) for large n, it provides better lower bounds for small values of n, 17 ≤ n ≤ 56; see Table 1.…”

“…The bound for n ≥ 62250 is due to Theorem 3. The bounds h 5 (12) ≤ 3, h 5 (13) ≤ 4, h 5 (14) ≤ 6, and h 5 (15) ≤ 9 are from [4,19].…”