happy-end problem prof. silvia fernández. happy-end problem posed by ester klein in the 1930s
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Happy-End Problem
Prof. Silvia Fernández
Happy-End Problem
Posed by Ester Klein in the 1930s.
gon?-convex a of
set vertex theforming points contains plane in the
position generalin points ofset any such that
number minimum theis what , 3Given
n
n
ng
ngn
Known and unknown results
Posed1930s, Ester Klein
First paper *1935, Erdős-Szekeres
Lower bounds Exact Upper bounds
* g(n) ≤ C(2n-4,n-2)+1**g(n) ≤ C(2n-4,n-2) ***g(n) ≤ C(2n-4,n-2)+7-2n****g(n) ≤ C(2n-5,n-2)+2
g(3)=3 trivialg(4)=5 Ester Kleing(5)=9 Klaibfleish, 1970g(n) open for all n ≥6
* 2n-2+1 ≤ g(n)
Conjecture. g(n)= 2n-2+1 for all n ≥6$500 “Erdős dollars”
**Chung, Graham, 1998***Kleitman, Patcher, 1998****Tóth, Valtr, 1998
First bounds (1935, Erdős-Szekeres)
12
42)(
n
nng 1
2
42)(
n
nng 1
2
42)(
n
nng
.12
42)(12
,3
2-n
n
nng
nall ForTheorem.
Lower bound(An example of 2n-2+1 points in general position with no convex n-gon.)
k-caps
Slopes of sides aredecreasing.(from left to right)
Lower bound(An example of 2n-2+1 points in general position with no convex n-gon.)
k-cups
Slopes of sides areincreasing.(from left to right)
Proof. By induction on p+q.
For p=1 the (q+1)-cap and has no 3-cup and no (q+2)-cap.
For q=1 the (p+1)-cup and has no (p+2)-cup and no 3-cap.
caps.-)2( no and cups-)2( nowith
points ofset a is here
qp
p
qpTLemma.
caps.-)2( no and cups-)2( nowith
points ofset a is here
qp
p
qpTLemma.
Proof. (Cont.)
Assume now that p≥2 and q≥2 and that there are sets
caps-)2( no and cups-)1( no with points 1
11
qpp
qpS
caps-)1( no and cups-)2( no with points 1
12
qpp
qpS
caps.-)2( no and cups-)2( nowith
points ofset a is here
qp
p
qpTLemma.
Proof. (Cont.) Construct the set S.
S1
S2S
.',',,:'
'max
21for is,That .within
slopes all and , within slopes allthan
largermuch is and overlap"not do"
and that soenough large be Let
2
1
21
kk SyxSyxxx
yym
,, kS
S
m
SSm
2, mm
caps.-)2( no and cups-)2( nowith
points ofset a is here
qp
p
qpTLemma.
Proof. (Cont.) Construct the set S.
possible. as
large as with in cup- a be
)},),...,,),,{(Let 2211
tSt
y(xy(xyxC tt
.1 then If 1 ptSC
.2 then If 2 ptSCS1
S2S
2, mm
caps.-)2( no and cups-)2( nowith
points ofset a is here
qp
p
qpTLemma.
caps.for argument Similar
. 2 and 1 then in cap-an is
since and 1 so and impossible iswhich
.
Note )}.,),...,,{(
and )},),...,,),,{(
then Assume
11
12
12
1
1
112
22111
ptpiSiSC
it
mxx
yy
xx
yym
y(xyxSC
y(xy(xyxSC
ii
ii
ii
ii
ttii
ii
Proof. (Cont.)
S1
S2
An example of 2n-2+1 points in general position with no convex n-gon.
Let P be a very large regular (4n-4)-gon centered at the origin. Consider the n-1 vertices of P that lie between the rays from the origin forming 45 and -45 degrees with the x-axis.
The example (Proof of lower bound)
1n
Q
The example (Proof of lower bound)
1 and 1-between are within slopes all
cap- no and cup-2 no has
points 2
has
such that ,, ... , , , sets heConsider t 2210
i
i
i
n
T
iniT
i
n-T
TTTT
The example (Proof of lower bound)
s. theofunion theis
., ... , , , sets by the of points selected 1 theReplace 2210
i
n
TQ
TTTTPn
0T1T
2nT
...
Q
The example (Proof of lower bound)
gons.-contain not does So
points. 11112imost at has Then
point. onemost at contains
cap. a is and cup a is
. and intersects such that index largest the andsmallest theLet
.in contained are verticesosepolygon whconvex a be Let
points. 22
2...
1
2
0
2 has
works. that prove weNow
2
nQ
nijjnC
TC
TCTC
TTCji
QC
n
nnnQ
Q
k
ji
ji
n
0T1T
2nT
...
Q
Upper bound
cap.-)2( aor cup-)2( a contains
position generalin points 1 ofset ny
qp
p
qpALemma.
cap.-)2( aor cup-)2( a contains
position generalin points 1 ofset ny
qp
p
qpALemma.
.12
42)(
n
nng
gon.-convex a containsposition
generalin points 12
42 ofset any Then
lemma. in the 2Let
n
n
n
nqp
bound. of Proof
cap.-)2( aor cup-)2( a contains
position generalin points 1 ofset ny
qp
p
qpALemma.
cap.-)2( aor cup-)2( a contains
position generalin points 1 ofset ny
qp
p
qpALemma.
gon.-convex a containsposition
generalin points 12
42 ofset any Then
lemma. in the 2Let
n
n
n
nqp
Proof.
Empty convex polygons
Posed in 1978.
?interior itsin of points nowith gon -convex a of
set vertex theforming points contains plane in the
position generalin points of set any such that
number minimum theis what , 3Given
Pn
n
nfP
nfn
Known and unknown resultsPosed
1978, Erdős
Known f(6)
f(3)=3 trivial f(4)=5 Easy f(5)=10 Hambort, 1978 f(n) does not exist for all n ≥7 Horton, 1983
f(6) open f(6)≥27 Overmars, 1989 best known
More questions
Rectilinear crossing number
points? ofset any
for guaranteecan that weralsquadrilate ofnumber
maximum theis what ral,quadrilateconvex a contains
points moreor 5 ofset any that know that weNow
n