Tur´an-type results for partial orders and
intersection graphs of convex sets
Jacob Fox
∗
J´anos Pach
†
Csaba D. T´oth
‡
Abstract
We prove Ramsey-type results for intersection graphs of geometric objects the plane. In particular,
we prove the following bounds, all of which are tight apart from the constant c. There is a constant
c > 0 such that for every family F of n ≥ 2 convex sets in the plane, the intersection graph of F or
its complement contains a balanced complete bipartite graph of size at least cn. There is a constant
c
0
> 0 such that for every family F of x-monotone curves in the plane, the intersection graph G of F
contains a balanced complete bipartite graph of size at least cn/ log n or the complement of G contains
a balanced complete bipartite graph of size at least cn. Our bounds rely on new Tur´an-type results on
incomparability graphs of partially ordered sets.
1 Introduction
A classic result of Erd˝os and Szekeres [10] in Ramsey theory states that every graph on n vertices contains a
clique or an independent set of size
1
at least
1
2
log n. This bound is tight up to a constant factor: Erd˝os [7]
showed that there exists a graph on n vertices, for every integer n > 1, with no clique or independent of
more than 2 log n vertices. Erd˝os and Hajnal [8] proved that certain graphs contain much larger cliques or
independent sets: For every hereditary family F of graphs other than the family of all graphs, there is a
constant c(F) > 0 such that every graph in F with n vertices contains a clique or an independent set of size
at least e
c(F)
√
log n
. (A family of graphs is hereditary if it is closed under taking induced subgraphs.) They
also asked whether this bound can be improved to n
c(F)
.
A complete bipartite graph, whose vertex classes are of the same size or their sizes differ by at most one,
is said to be balanced. A balanced complete bipartite graph with n vertices is called a bi-clique of size n.
The problem of Erd˝os and Hajnal motivates the definition of the following two properties of a hereditary
family F of graphs: We say that
1. F has the Erd˝os-Hajnal property if there is a constant c(F) > 0 such that every graph in F on n
vertices contains a clique (that is, a complete subgraph) or an independent set of size n
c(F)
.
2. F has the strong Erd˝os-Hajnal property if there is a positive constant b(F) such that every graph G ∈ F
with n > 1 vertices or its complement G contains a bi-clique of size b(F)n.
Alon et al. [1] proved that the strong Erd˝os-Hajnal property implies the Erd˝os-Hajnal property. For partial
results on the Erd˝os-Hajnal problem, see [2], [3], [4], and [9].
The intersection graph of a set system is a graph whose vertices are in one-to-one correspondence with
the sets, with two vertices being connected by an edge if and only if the corresponding sets have at least
∗
Department of Mathematics, Princeton University, Princeton, NJ 08544. Email: jacobfox@math.princeton.edu. Supported
by NSF Graduate Research Fellowship and a Princeton Centennial Fellowship.
†
City College, CUNY and Courant Institute, NYU, New York, NY, USA. Email: pach@cims.nyu.edu. Supported by NSF
Grant CCF-05-14079, and by grants from NSA, PSC-CUNY, Hungarian Research Foundation OTKA, and BSF.
‡
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA. Email: toth@math.mit.edu
1
All logarithms in this paper are of base two.
1
one element in common. As noted by Ehrlich, Even, and Tarjan [6], not every graph can be realized as the
intersection graph of connected sets in the plane. For instance, the bipartite graph on 15 vertices formed by
replacing each edge of K
5
by a path of length 2 has no such realization. This implies, using the above result
of Erd˝os and Hajnal, that the intersection graph of any n connected sets in the plane contains a clique or
an independent set of size e
c
√
log n
, for some absolute constant c > 0. This general bound has been improved
for families of intersection graphs of certain geometric objects in the plane.
Pach and Solymosi [16] proved that the family of intersection graphs of line segments in the plane has
the strong Erd˝os-Hajnal property. Later, Alon et al. [1] generalized this result to intersection graphs of
d-dimensional semialgebraic sets of degree at most D, for any fixed positive integers d and D.
In this paper, we prove similar results for intersection graphs of convex sets and x-monotone curves (that
is, continuous curves in the plane such that every line parallel to the y-axis intersects them in at most
one point). A common feature of these objects is that the boundaries of two convex sets, as well as two
x-monotone curves, may intersect in an arbitrary number of points, in sharp contrast to semialgebraic sets
in “general position.”
Theorem 1 The family of intersection graphs of convex sets in the plane has the strong Erd˝os-Hajnal
property. That is, there exists a constant c > 0 with the property that the intersection graph G of any
collection of n > 1 convex sets contains a bi-clique of size cn, or its complement G contains a bi-clique of
size cn.
The (weak) Erd˝os-Hajnal property for the family of intersection graphs of compact convex sets in the
plane has been established by Larman et al. [15, 18]. For the bipartite version, the best previous result [12]
was that the intersection graph G of any collection of n compact convex sets in the plane, or its complement
G, contains a bi-clique of size n
1−o(1)
.
Theorem 1 does not generalize to higher dimensions: Tietze [21] showed that every graph can be realized
as the intersection graph of convex compact sets in R
3
.
Theorem 2 There exists a constant c > 0 with the property that the intersection graph G of any collection
of n > 1 x-monotone curves in the plane satisfies at least one of the following two conditions:
(a) G contains a bi-clique of size
cn
log n
; or
(b) G, the complement of G, contains a bi-clique of size cn.
The last theorem easily generalizes to vertically convex objects, that is, to connected sets with the property
that every vertical line intersects them in a connected interval, which may consist of just one point or may
be empty. To see this, notice that for every finite collection of vertically convex objects in the plane, one
can construct a collection of x-monotone curves with the same intersection graph: Pick a “witness” point in
the intersection of each intersecting pair of objects, and within each object connect all witness points by a
vertically convex curve. Slightly perturbing the picture, if necessary, we can ensure that none of these curves
contains a whole vertical segment, that is, the curves are x-monotone.
The comparability graph (incomparability graph) of a partially ordered set, in short, poset, (P, ≺) is a
graph defined on the vertex set P so that two elements of P are adjacent if and only if they are comparable
(incomparable). Every partially ordered set is the intersection of its linear extensions. The dimension of a
poset is the minimum number of its linear extensions whose intersection is that poset.
One may wonder whether condition (a) in Theorem 2 can be replaced by the stronger property that G
contains a bi-clique of size cn. This is not the case: It is easy to check [17, 19] that every incomparability
graph is isomorphic to the intersection graph of x-monotone curves (in fact, continuous real functions defined
on [0, 1]). Using this observation, a construction of Fox [11] shows that Theorem 2 is the best possible.
The proofs of Theorems 1 and 2 crucially depend on Tur´an-type results for incomparability and com-
parability graphs. Tur´an’s classic problem is to determine ex(n, H), the maximum number of edges that a
graph with n vertices can have without containing a (not necessarily induced) subgraph isomorphic to H.
Let C and I denote the families of comparability graphs and incomparability graphs. For any d, let C
d
and I
d
denote the families of comparability graphs and incomparability graphs of dimension d. Furthermore,
2
let
ex
C
(n, G) = max{|E(G)| : G ∈ C, H 6⊆ G, and |V (G)| = n},
and define the functions ex
C
d
(n, G), ex
I
(n, G), and ex
I
d
(n, G) analogously.
If the excluded graph H is a clique, according to Tur´an’s theorem [22], ex(n, K
t
) is attained for the
balanced complete (t − 1)-partite graph with n vertices. Since every (t − 1)-partite complete graph is both
a comparability graph and an incomparability graph, we obtain that ex(n, K
t
) = ex
C
(n, K
t
) = ex
I
(n, K
t
),
for all n, t ≥ 2.
On the other hand, if the excluded graph is a bi-clique, Tur´an’s questions, when restricted to comparability
and incomparability graphs, have very different answers than the “unrestricted” versions.
In Section 2, we establish the following two results, needed for the proofs of Theorems 1 and 2.
Theorem 3 The maximum number of edges of a K
t,t
-free (in)comparability graph of a 2-dimensional poset
with n elements satisfies
ex
I
2
(n, K
t,t
) = ex
C
2
(n, K
t,t
) ≤ 2(t − 1)n −
µ
2t − 1
2
¶
,
for every t ≥ 2 and n ≥ 2t − 1.
Theorem 4 There is a constant c > 0 such that for every δ > 0 and n ∈ N, we have
ex
I
(n, K
t,t
) < δn
2
, where t =
¹
cδn
log
1
δ
log n
º
.
In other words, if a poset P on n vertices has at least δn
2
incomparable pairs, then its incomparability graph
contains a bi-clique of size Ω(δn/(log
1
δ
log n)). Note that the size of the largest bi-clique in a random graph
with n vertices and δn
2
edges (and in its complement) is almost surely O
δ
(log n), for any 0 < δ < 1.
In Section 3, we establish an analogue of Theorem 4 for comparability graphs of posets (Theorem 7).
It will not be needed for the pro of of Theorems 1 and 2, but it will enable us to strengthen a theorem of
Fox [11] (see Theorem 8).
It is very easy to see that it is sufficient to establish Theorems 1 and 2 for collections of sets intersecting the
same line. To deal with such collections, in Sections 4 and 5 we develop some auxiliary results (Lemmas 10, 13
, and 14) for “flags” and “bridges,” that is, for connected sets that are incident to one line or lie between two
parallel lines, respectively. One is designed to address the case when the average degree of the vertices in the
intersection graph G is smaller than ε|V (G)|, for a suitable constant ε ∈ (0, 1), while the other two analyze
the opposite situation. In the first case, we show the existence of a large bi-clique in the complement of G,
and in the latter ones, in G itself. In these latter cases, we use the Tur´an-type results for incomparability
graphs, established in Section 2.
The pieces of the proofs of Theorems 1 and 2, following the above strategy, are put together in Section 6.
The last section contains a few remarks and open problems.
2 Tur´an-type results for incomparability graphs
The aim of this section is to prove Theorems 3 and 4.
Let ex
I
d
(n, K
t,t
) (and ex
C
d
(n, K
t,t
)) be the maximum number of edges that a K
t,t
-free graph of n vertices
can have if it is the comparability (incomparability, resp.) graph of a d-dimensional partial order. We call a
graph G r-degenerate if every subgraph of G contains a vertex of degree at most r. Clearly, the number of
edges of any r-degenerate graph G with n > r vertices satisfies
|E(G)| ≤ rn −
µ
r + 1
2
¶
,
and this bound is tight.
3
In the sequel, we use the notation [n] = {1, . . . , n}. For any permutation π of [n], let P
π
= ([n], <
π
)
denote the 2-dimensional partial order on [n], in which i <
π
j if and only if i < j and π(i) < π(j).
Proof of Theorem 3: It is sufficient to prove the statement for incomparability graphs, because the
corresponding statement for comparability graphs follows by the simple observation that C
2
= I
2
. Further,
it is enough to show that the incomparability graph of every 2-dimensional partial order is (2t−2)-degenerate.
Every 2-dimensional poset of n vertices can be realized as P
π
, for a suitable permutation π. Suppose for
contradiction that the degree of every vertex of the incomparability graph P
π
is at least 2t −1. Notice that
every i ∈ [n] is incomparable with at most i − 1 + π(i) − 1 other elements of [n]. Since each element i ∈ [n]
is incomparable with at least 2t − 1 other elements of P
π
, we have π(i) ≥ t + 1 for i ∈ [t] and i ≥ t + 1
for π(i) ∈ [t]. In particular, every i ∈ [t] is incomparable with every element j with π(j) ∈ [t]. Hence, the
incomparability graph contains K
t,t
, which is a contradiction. 2
The bound in Theorem 3 is roughly within a factor of two from the truth. To see this, consider the
following simple construction. Let n = 2`(t − 1), for some ` ∈ N, and let P
π
denote the 2-dimensional poset
defined by the permutation π(i + 2k(t − 1)) = 2t − i + 2k(t − 1), for 1 ≤ i ≤ 2t − 1 and 0 ≤ k ≤ ` − 1.
The incomparability graph of P
π
is the disjoint union of cliques of size 2t − 1, hence it is K
t,t
-free and
(2t − 2)-regular, so that its number of edges is (t − 1)n.
Corollary 5 For all positive integers d, n, and t with n ≥ 2t − 1, we have
ex
I
d
(n, K
t,t
) ≤ (d − 1)
µ
2(t − 1)n −
µ
2t − 1
2
¶¶
.
Proof: Consider a d-dimensional poset (P, ≺) whose incomparability graph does not contain K
t,t
as a
subgraph. We may assume that P = [n] and that there are permutations π
1
, . . . , π
d
of P with π
1
being the
identity permutation such that i ≺ j if and only if π
k
(i) < π
k
(j) for every k ∈ [d].
Two elements, i and j with i < j, are incomparable if and only if there is an index k ∈ [2, d] such
that π
k
(i) > π
k
(j). Hence, the number of edges of the incomparability graph of (P, ≺) is at most the
sum of the number of edges in the d − 1 incomparability graphs of the 2-dimensional partially ordered sets
P
π
2
, . . . , P
π
d
. For k ∈ [2, d], the incomparability graph of P
π
k
does not contain K
t,t
, since otherwise the
incomparability graph of (P, ≺) contains K
t,t
. By Theorem 3, the incomparability graph of (P, ≺) has at
most (d − 1)
¡
2(t − 1) n −
¡
2t−1
2
¢¢
edges. 2
It is a simple corollary to Dilworth’ theorem [5] that every partially ordered set on n elements contains
a chain or an antichain of size at least
√
n. For the proof of Theorem 4, we need the following bipartite
analogue of this result.
Lemma 6 (Fox [11]) If n is sufficiently large, then every poset of n elements contains two disjoint subsets
A and B, each of size at least
n
4 log n
, such that either every element of A is larger than every element of B
or every element of A is incomparable with every element of B.
Given a poset (P, <), for any x ∈ P and for any subset S ⊆ P , define D
S
(x), the down-set of x in S, as
the set of elements in S below x. That is, let D
S
(x) = {s : s ∈ S and s < x}. Analogously, let the up-set of
x in S be defined as U
S
(x) = {s : s ∈ S and s > x}.
Proof of Theorem 4: Let (P, <) be a poset with n elements, whose incomparability graph contains no
K
t,t
. Let <
∗
be a linear extension of <, let X and Y denote the set of the top b
n
2
c and the set of the bottom
d
n
2
e elements of P with respect to <
∗
. Clearly, we have P = X ∪ Y . Let X
1
be the set of all x ∈ X with
|D
X
(x)| ≥ t, and let X
2
= X \ X
1
be its complement. Similarly, let Y
1
be the subset of all y ∈ Y with
|U
Y
(y)| ≥ t, and let Y
2
= Y \ Y
1
.
Every x ∈ X
1
is comparable with every y ∈ Y
1
, otherwise every element of D
X
(x) is incomparable with
every element of U
Y
(y), which means that the incomparability graph of (D
X
(x)∪D
Y
(y), <) already contains
a K
t,t
.
4
For every x ∈ X
2
, the down-set of x in X is smaller than t, and so the comparability graph of (X
2
, <)
contains no K
t,t
. By Lemma 6, however, the comparability graph or the incomparability graph of (X
2
, <)
contains K
s,s
with s =
|X
2
|
4 log |X
2
|
, provided that |X
2
| is sufficiently large. Hence, t >
|X
2
|
4 log |X
2
|
if |X
2
| is
sufficiently large; and likewise, t >
|Y
2
|
4 log |Y
2
|
if |Y
2
| is sufficiently large. Since log | X
2
| ≤ log n and log |Y
2
| <
log n, it follows that for every sufficiently large n, we have |X
2
| ≤ 4t log n and |Y
2
| ≤ 4t log n. Every element
of P is incomparable with at most n − 1 other elements of P , and so, for n sufficiently large, the elements
of X
2
and Y
2
participate in at most 8t(n −1) log n incomparable pairs of P . Since there is no incomparable
pair (x, y) with x ∈ X
1
and y ∈ Y
1
, we have
ex
I
(n, K
t,t
) ≤ 8t(n − 1) log n + 2ex
I
³l
n
2
m
, K
t,t
´
,
if n is large enough. Using the fact that every graph with m vertices has at most
¡
m
2
¢
edges, after iterating
the above inequality j times, we obtain
ex
I
(n, K
t,t
) = O (jtn log n) +
n
2
2
j
.
Setting j := dlog
1
δ
e + 1, the theorem follows. 2
3 Tur´an-type results for comparability graphs
In this section, we estimate the function ex
C
(n, K
t,t
), the maximum number of edges that a K
t,t
-free com-
parability graph with n vertices can have.
Instead of studying ex
C
(n, K
t,t
) directly, it will be more convenient to bound its inverse. Let T (n, m)
denote the largest integer t such that every comparability graph with n vertices and at least m edges contains
K
t,t
. The following theorem demonstrates a dramatic change in T (n, m), when m is roughly n
2
/4.
Theorem 7 (1) For every ε > 0, there is a constant c(ε) such that T (n, (
1
4
− ε)n
2
) ≤ c(ε) log n.
(2) There are constants c
1
> 0 and c
2
> 0 such that c
1
√
n ≤ T (n,
n
2
4
) ≤ c
2
√
n log n.
(3) For every ε > 0, we have T (n, (
1
4
+ ε)n
2
) ≥
ε
2
n.
Proof: We first prove the upper bounds on T (n, m). Note that every bipartite graph is a comparability
graph. Consider a random bipartite graph with d
n
2
e vertices in the first class and b
n
2
c vertices in the
second class, there is an edge between any two vertices independently at random with probability p. Letting
p = 1 − ε, it is an easy exercise to show that with positive probability this random bipartite graph has at
least (
1
4
− ε)n
2
edges and contains no bi-clique of size c(ε) log n for some constant c(ε). This proves (1).
Let P be the p oset on n > 1 elements which has d
√
n log ne elements that form a chain and are larger
than the n − d
√
n log ne remaining elements, and the comparability graph of the remaining n − d
√
n log ne
elements is a random bipartite graph with at least b
n−d
√
n log ne
2
c elements in each of its classes and each
edge taken randomly with probability p = 1 −
q
log n
n
. It is easy to check that with positive probability, the
comparability graph G of the poset P has at least n
2
/4 edges and the largest bi-clique in G has O(
√
n log n)
vertices. This establishes the upper bound in (2).
We next prove the lower bounds on T (n, m). Let P be a partially ordered set such that its comparability
graph G does not contain K
t,t
. Let X be the subset of P consisting of the elements x ∈ X with |D
P
(x)| ≥ t,
and let Y be the subset of P consisting of the elements y ∈ Y with |U
P
(y)| ≥ t. Let Z denote P \ (X ∪ Y ).
The sets X and Y are disjoint, for if x ∈ X ∩ Y , then U
P
(x) and D
P
(x) each have at least t elements and
every element of U
P
(x) is larger than every element of D
P
(x), a contradiction. Hence, P = X ∪ Y ∪ Z is a
partition of P . Every element of Z belongs to at most 2 t − 2 comparable pairs. Every element x ∈ X has
5
