F. R. K. Chung y. University ofpennsylvania. Philadelphia, Pennsylvania R. L. Graham. AT&T Labs - Research. March 2,1997.

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1 Forced convex n-gons in the plne F. R. K. Chung y University ofpennsylvni Phildelphi, Pennsylvni R. L. Grhm AT&T Ls - Reserch Murry Hill, New Jersey Mrch 2,1997 Astrct In seminl pper from 1935, Erd}os nd Szekeres showed tht for ech n there exists lest vlue g(n) such tht ny suset of g(n) points in the plne in generl position must lwys contin the vertices of convex n-gon. In prticulr, they otined the ounds 2 +1 g(n) 2n 4 n 2 +1 which hve stood unchnged since then. n 4. In this note we remove the +1 from the upper ound for 1. The min result In 1935, Pul Erd}os nd George Szekeres pulished short pper \A comintoril prolem in geometry" [1] which ws destined to hve profound inuence on the development of comintorics (nd especilly Rmsey theory) during the next 60 yers (cf. [3]). In prticulr, in this pper Erd}os nd Szekeres rediscovered Rmsey's theorem, which hd only just ppered (unknown to them) ve yers erlier. Their investigtions rose from geometricl question of the tlented young mthemticin Esther Klein (soon to ecome Mrs. Szekeres). She sked, \Is it true tht for every n, there is lest vlue g(n) such tht ny set X of g(n) points in the plne in generl position lwys contins the vertices of convex n-gon?" Erd}os nd Szekeres gve severl proofs of the existence of g(n) in [1] nd estlished the following ounds:! 2 +1 g(n) 2n 4 +1: (1) n 2 They lso conjectured tht the lower ound in (1) in fct lwys holds with equlity. This is known [2] to e the cse for n 5. Despite repeted ttempts over the yers, no generl improvement on (1) hs een found. In this note, we mke very smll improvement on the upper ound of (1). Nmely, we show for n 4. g(n) 2n 4 n 2 To pper in Discrete nd Computtionl Geometry y Reserch supported in prt y NSF Grnt No. DMS ! (2)

2 While this is dmittedly rther modest, we hope tht it might suggest methods which could give rise to more sustntil reductions in the upper ound. Proof of (2): By n m-cp we men sequence of m points x 1, x 2 ::: x m such tht the polygonl pth connecting them is concve, i.e., the x i hve incresing x-coordintes nd the pth from x 1 to x m turns clockwise t ech intermedite vertex. Similrly, nm-cup is set of points y 1 y 2 ::: y m with incresing x-coordintes such tht the polygonl pth joining them is convex, i.e., the pth from y 1 to y m lwys turns counterclockwise. 5 - cp 6-cup Figure 1: Cps nd cups The following result from [1] follows esily y induction. Lemm 1. If X E 2 -cp or -cup. is in generl position nd jxj > +4 2 then X contins either n In fct, s shown in [1], this ound is shrp. Theorem If X E 2 is in generl position nd jxj 2n4 for n 4, thenx contins the vertices of convex n-gon. Proof: Suppose the contrry. Rotte X if necessry so tht no line determined y two points of X is either horizontl or verticl. We cn further ssume without loss of generlity tht ll lines determined y two points of X hve slopes less thn 0.1 in solute vlue (y uniformly compressing X in the y-direction, if necessry). Dene A := fx 2 X : x is the left-hnd endpoint of some (n 1)-cp in Xg. 2n5 Cse 1. jaj > n3. Then y Lemm1,A contins n (n 1)-cup, sy, y 1 y 2 ::: y n1. Since y n1 2 A, there y n1 = z 1 z 2 y 1 y z n1 y 2 Figure 2: A cup joining cp exists n (n 1)-cp y n1 = z 1 z 2 ::: z n1 in X. However, this is impossile since either y 1, y 2 :::y n1 z 2 is n n-cup, or y z 1 z 2 ::: z n1 is n n-cp (see Fig. 2). 2n5 Cse 2. jaj < n3. 2

3 Then B := X n A stises 2n4 jbj > contrdiction. This leves s the only possiility: Cse 3. jaj = jbj = 2n5 n3 = n4 2n5 n3 = 2n5 n3 nd, similrly s in Cse 2, we rech For ny 2 B, consider the set A [fg. Since this set hs size greter thn 2n5 n3 then y Lemm 1, it contins n (n 1)-cup, sy with right-hnd endpoint y. Now, if y 2 A then s in Cse 1, we rech contrdiction. Hence we must hve y =. Thus, ech 2 B is the right-hnd endpoint of n(n 1)-cup with left-hnd endpoint in A. It follows in similr wy tht ech 2 A is the left endpoint ofn(n 1)-cp with right-hnd endpoint in B. We now form directed iprtite grph G with vertex sets A nd B, nd edge set E consisting of ll pirs (u v), where either u 2 A is the left-hnd endpoint nd v 2 B is the right-hnd endpoint of some (n 1)-cp in X, orv 2 A is the left-hnd endpoint nd u 2 B is the right-hnd endpoint of some(n 1)-cup in X. By the preceding remrks, it follows tht ll vertices of G hve outdegree t lest one. This implies G hs some (directed) cycle C = i1 i1 ir ir. Now consider n edge ( ) 2 E. Let L + () denote the hlf-line strting t nd going down with slope 0.1, nd let R () denote the hlf line strting t nd going down with slope 0:1. Also, let S( ) denote the line segment joining nd. Finlly, lety ( ) denote the region of E 2 (strictly) elow the pth L + ()S( )R () (see Fig. 3). (n 1)-cp S( ) R () L + () Y ( ) Figure 3: Clim 1. X hs no point in Y ( ). Otherwise, if x 2 X \ Y ( ) then the (n 1)-cp spnned y ( ) together with x forms convex n-gon in X, which is contrdiction. 3

4 By n nlogous rgument for ( ) 2 E, withl (), R + (), Y ( ) dened ccordingly (see Fig. 4), we lso see tht Y ( ) cn contin no point ofx. Y ( ) L () S( ) R + () (n 1)-cup Figure 4: Next, consider two connected edges ( ) nd ( 0 )ine. We cnnot hve = 0, since if we did, then X would contin convex (2n 4)-gon (formed y the (n 1)-cp nd (n 1)-cup spnned y nd ), which is impossile. Clim 2. 0 must lie ove the line through nd. Proof: Suppose not. Then from the geometry of the sitution (see Fig. 5), either 0 2 Y ( ) or 2 Y ( 0 ),contrdiction. A similr rgument shows if ( ) 2 E nd ( 0 ) 2 E then 0 Y ( 0 ) 0 0 Y ( ) Figure 5: must lie elow the line through nd. Finlly, consider the cycle C = i1 i1 ir ir occurring in G. Ifr =1thenwend convex (2n 4)-gon, which is impossile. So, we my ssume r 2. By Clim 2, ech of the ngles etween djcent edges, i1 i1 i1 i2 i2 i2 ir ir ir i1 must turn in counterclockwise direction. Hence, the lines through the consecutive edges i1 i1 i1 i2 i2 i2 hve incresing slopes, nd ny pir of these lines intersects t n ngle of less thn 2 rctn 0:1 < 12. However, since ll of the slopes of the lines re etween 0:1 nd 0.1, nd C is cycle, we rech contrdiction. We re inclined to elieve (s did Erd}os nd Szekeres) tht the lower ound 2 +1is the true vlue of g(n). However, we dmit tht there is little rel evidence yet for this elief. 4

5 References [1] P. Erd}os nd G. Szekeres, A comintoril prolem in geometry, Compositio Mth. 2 (1935), [2] P. Erd}os nd G. Szekeres, On some extremum prolems in elementry geometry, Ann. Univ. Sci. Budpest. Eotvos Sect. Mth. 3{4 (1961), 53{62. [3] R. L. Grhm nd J. Nesetril, Rmsey theory in the work of Pul Erd}os, The Mthemtics of Pul Erd}os, (R. L. Grhm nd J. Nesetril, eds.), Springer Verlg, Heidelerg,