1. Complete bipartite graphs
1.1. K¨ovari-S´os-Tur´an theorem. When H is a bipartite graph, i.e., when χ(H) = 2, Erd˝os and Stone’s theorem asserts that πH = 0. In other words,
for each fixed positive real ε, there exists n0 such that for n > n0, if G is an
n-vertex graph with at least εn2 edges, then G contains H as a subgraph. Thus bipartite graphs are ‘degenerate cases’ in extremal problems for graphs. We prove the following much stronger result, due essentially to K˝ovari, S´os, and Tur´an. (Note that for every bipartite graph H, there exists s and t such that H ⊆ Ks,t).
Theorem 1. (K˝ovari, S´os, Tur´an) For any natural numbers s and t with s ≤ t, there exists a constant c such that
ex(n, Ks,t) ≤ cn2−1/s.
Proof. Suppose that G is an n-vertex graph with at least cn2−1/s edges. Count the pairs (v, S) consisting of a vertex v ∈ V (G) and a set S ⊆ N (v) of size |S| = s. The number of such pairs is
X v d(v) s ≥ n 1 n P vd(v) s ≥ n2cn 1−1/s s ≥ nc sns−1 s! = c sns s! for n sufficiently large. If cs > t − 1, then there exists a set S that is counted at least t times in the equation above. This gives a copy of Ks,t in G.
A more careful analysis shows that the following bound holds: ex(n, Ks,t) ≤ (1 + o(1))
1 2(t − 1)
1/sn2−1/s.
(1)
It is known that this bound is sharp in some specific cases. We will come back to this later. We present a quick application of K˝ovari-S´os-Tur´an theorem to the unit distance problem of Erd˝os.
Theorem 2. A set of n points in R2 determines at most cn3/2 unit dis-tances.
Proof. Let X be a given set of n points in R2. Let G be a graph on the
vertex set X where two points x, y are adjacent if and only if |x − y| = 1. Note that G is K2,3-free. Therefore by K˝ovari-S´os-Tur´an theorem, G has at
most cn3/2 edges.
The best known bound to the problem is O(n4/3) first proven by Sze-mer´edi and Trotter. By now there are several different proofs of the same bound. Any improvement on the exponent would be extremely interesting. It is conjectured that the bound should be n1+o(1). The construction that motivated this bound is the√n ×√n grid. If we re-scale the grid so that the most popular distance is 1, then there are about n1+c/ log log n unit distances in this construction.
1.2. Random Constructions. The binomial random graph G(n, p) is the probability space of graphs with vertex set [n] where each pair of vertices form an edge independently with probability p. We say that G(n, p) has a property P asymptotically almost surely if the probability that G(n, p) has P tends to 1 as n tends to infinity.
Here we examine the values of p for which G(n, p) contains Ks,s. Let N be
the random variable counting the number of copies of Ks,s. For a fixed pair
of sets X, Y of sizes |X| = |Y | = s, the probability that the bipartite graph between X and Y is complete is ps2. Therefore by linearity of expectation, we have E[N ] ≈ n 2s 2(s!)2 · p s2 .
If n2sps2 = (n2ps)s< 12 (this happens when p ∼ n−2/s), then since 1
2 > E[N ] ≥ P(N ≥ 1),
we know that N = 0 with probability at least 12. Furthermore, it is well known that in this range of p, the number of edges in G(n, p) is a.a.s. n2p ∼ n2−2/s.
One can slightly improve this construction. Note that if N ≤ n42p, then we can remove one edge from each copy of Ks,s to obtain a Ks,s-free graph
on n vertices with at least (about) n42p edges. Note that E[N ] < n42p occurs when p ∼ n−2/(s+1). Therefore for all s ≥ 2,
c1n2−2/(s+1)≤ ex(n, Ks,s) ≤ c2n2−1/s.
1.3. Algebraic Constructions. The following elegant construction, first discovered by Erd˝os, R´enyi, and S´os in 1966, shows that the bound (??) is tight for s = t = 2 for infinitely many values of n. For a prime power q, we denote the finite field of order q by Fq.
Theorem 3. (Erd˝os, R´enyi, and S´os) For infinitely many values of n, there exists a graph on n vertices and about n3/2 edges where each pair of vertices have exactly one common neighbor.
Proof. Let p be a prime. Define a equivalence relation over F3p where two
points (a1, a2, a3), (a10, a02, a03) are equivalent if and only if there exists λ ∈ Fp
the equivalence class that contains (a, b, c) (this space is called a projective plane).
Let V = {[a, b, c] : a, b, c ∈ Fp, (a, b, c) 6= (0, 0, 0)} and note that |V | = p3−1
p−1 = p
2+ p + 1. Let G be a graph over vertex set V where [a, b, c] and
[a0, b0, c0] are adjacent if and only if aa0 + bb0 + cc0 = 0. We claim that each pair of vertices in G have exactly one common neighbor. Indeed, let v = [a, b, c] and v0 = [a0, b0, c0] be two distinct vertices of G. Then [x, y, z] can be adjacent to both vertices only if
ax + by + cz = 0 and a0x + b0y + c0z = 0.
Since [a, b, c] and [a0, b0, c0] are distinct vertices, the set of vectors (x, y, z) ∈ F3p satisfying both equations above forms a line in F3p. This means that the
non-zero solutions define a single equivalence class. Hence there exists a
unique common neighbor of v and v0.
For the diagonal cases s = t, the only other case where (??) is known to give the correct order of magnitude for ex(n, Ks,s) is when s = t = 3. Brown
considered the graph over F3p where two vertices (a, b, c) and (a0, b0, c0) are
adjacent if and only if (a − a0)2+ (b − b0)2+ (c − c0)2= 1, and showed that this is K3,3-free. For s ≥ 4, the diagonal case is not known but when t is
much larger than s, the bound is known to be tight up to the constant. This was first proved by Koll´ar, R´onyai, and Szab´o for t ≥ s! + 1, and was then improved by Alon, R´onyai, and Szab´o to t ≥ (s − 1)! + 1.
All the above mentioned constructions can be put into the following frame-work. Let V1 and V2 be two copies of Fkp and f be some polynomial over
Fp in 2k variables. Each case was based on some ‘clever’ polynomial f .
Re-cently, Bukh showed some interesting family of constructions by considering ‘random’ polynomials f instead.
2. Trees
In this section, we investigate the Tur´an number of trees and even cycles. The following lemma is another extremely useful tool in extremal graph theory.
Lemma 4. Let G be an n-vertex graph with at least nd edges. Then there exists a subgraph of minimum degree at least d + 1.
Proof. Repeatedly remove vertices of degree at most d from the graph. If we stop before and there exists at least 2 vertices, then we found a subgraph of minimum degree at least d + 1. Otherwise, we removed n − 1 vertices and there are no remaining edges. Therefore e(G) ≤ (n−1)d and this contradicts
the given condition.
Let T be a tree on t + 1 vertices. One can easily prove that t − 1
where the lower bound is for n divisible by t. The lower bound follows from considering a graph consisting of vertex disjoint copies of Kt. The upper
bound follows from Lemma ?? since if G is an n-vertex graph with at least (t − 1)n edges, then it contains a graph of minimum degree at least t into which we can greedily embed any tree on t + 1 vertices.
Hence the extremal number for trees is linear in the number of vertices. A famous conjecture of Erd˝os and S´os asserts that the lower bound is the right answer for all trees T . Recently, Ajtai, Koml´os, Simonovits, and Szemeredi announced a positive solution to this conjecture for sufficiently large n.
3. Even cycles
The following bound on the extremal number for even cycles was proved by Bondy and Simonovits.
Theorem 5. For each fixed positive integer t ≥ 2, there exists a constant c such that
ex(n, C2t) ≤ cn1+ 1 t.
By Lemma ??, we can find a subgraph G0 of minimum degree at least cn1/t. Fix an arbitrary vertex v of G0 and consider a breadth-first-search tree T constructed by starting the exploration at v. For each ` ∈ [0, t], let L` be the set of vertices at the `-th level of T . If all we wanted was a cycle of
length at most 2t (instead of exactly 2t), then we can easily finish the proof since for each ` ≤ t − 1, the inequality |L`+1| ≥ cn1/t|L`| holds, showing that
|Lt| ≥ ctn, and for c ≥ 1 contradicts the fact that G0 has at most n vertices.
However, it requires a considerable amount of work to find a cycle of length exactly 2t. A similar phenomenon occurs frequently in the study of extremal number of bipartite graphs, where one can easily prove the existence of a non-injective copy but is a lot more difficult to find an injective copy.
Theorem ?? is known to be the correct order of magnitude only for t = 2, 3, 5. Note that for t = 2, C4 is the same as K2,2. Hence the construction
given above shows the tightness for t = 2. For t = 3 and 5, the matching lower bound were found by Benson, and Singleton respectively. Recently, using the framework developed by Bukh mentioned in a previous section, Conlon made an interesting breakthrough. He showed that for every t, there exists an s such that for infinitely many values of n, there exists an n-vertex graph in which there are at most s paths of length t between each pair of vertices.
4. Sparse bipartite graphs
The following theorem extends K¨ovari-S´os-Tur´an theorem.
Theorem 6. (F¨uredi) There exists a constant c such that the following holds. Let H be a bipartite graph with bipartition A ∪ B where each vertex a ∈ A has at most d neighbors in B. Then for sufficiently large n,
We prove this theorem using dependent random choice.
Lemma 7. Suppose that G is a graph on n vertices with at least αn2 edges. Then for any integer r ≥ 1, there exists a subset of vertices A of size |A| ≥
1 2α
rn such that all r-tuple of vertices in A have at least 1 2αn
1/r common
neighbors.
Proof. Let v1, · · · , vrbe vertices chosen uniformly and independently at
ran-dom, and define A =Tr
i=1N (vi). Note that x ∈ A if and only if v1, · · · , vr
are all chosen from N (a). Hence P(x ∈ A) = |N (x)|n r. Therefore by the linearity of expectation and convexity,
E[|A|] = X x∈V (G) |N (x)| n r ≥ nαr.
Let M be the number of r-tuple of vertices in A with less than 12αn1/r com-mon neighbors. Note that for a fixed r-tuple of vertices R, the probability that R ⊆ A isd(R)n r, where d(R) is the number of common neighbors of R. Therefore by linearity of expectation,
E[M ] ≤ n r r! αn1/r−1 2 !r = α r r!2rn. Therefore E[|A| − M ] ≥ nα2r.
There exists a particular choice of v1, · · · , vr, for which |A| − M ≥ nα r 2 .
For this choice, let A0 be the set obtained from A by removing one vertex from each r-tuple with less than 12αn1/r common neighbors. Note that |A0| ≥ |A| − M ≥ nα2r, and all r-tuple of vertices in A0 have at least 12αn1/r
common neighbors.
The proof of Theorem ?? easily follows.
Proof of Theorem ??. Suppose that H is a bipartite graph with bipartition X ∪ Y , where |X ∪ Y | = h and all vertices in X have degree at most d. Let G be graph with n vertices and at least 2hn2−1/d edges. Apply Lemma ?? with α = 2hn−1/d and r = d to find a set A of size |A| ≥ 12cd≥ h such that all d-tuple of vertices in A have at least 12c ≥ h common neighbors.
We can now easily find a copy of H by a greedy algorithm. First arbitrarily embed X into A and call this map f . For each vertex y ∈ Y , note that f (N (y)) has at least h common neighbors. Since |X ∪ Y | = h, we can find one vertex in the common neighborhood that is not an image of f . Define f (y) as this vertex and repeat. In the end, we obtain a copy of H. A graph is d-degnerate if all its subgraphs have a vertex of degree at most d. Degeneracy is a natural measure of sparseness of a graph. For example if a graph is d-degenerate then every subset of vertices X contains at most |X|d edges. Also, there exists a linear ordering of the vertices so that each
vertex has at most d neighbors that precede itself. Erd˝os made the following conjecture in 1967.
Conjecture 8. (Erd˝os) For every d-degenerate bipartite graph H, ex(n, H) = O(n2−1/d).
The best known result towards this conjecture is the following theorem of Alon, Krivelevich, and Sudakov. Its proof is also based on dependent random choice.
Theorem 9. (Alon-Krivelevich-Sudakov) There exists a constant c such that the following holds. If H is a d-degenerate bipartite graph then for sufficiently large n,
ex(n, H) ≤ cn2−1/(4d).
At one point Erd˝os made the following conjecture which seems plausible given all the results mentioned above.
Conjecture 10. (Erd˝os) For any graph bipartite graph H, there exists α = α(H) such that limn→∞ex(n,H)nα exists. Furthermore, perhaps α = 1 + 1k or
α = 2 −k1.
The second part was disproved by Erd˝os and Simonovits. They showed (among other results) that there is a graph whose extremal number is Ω(n2−8/17) and O(n8/5). (the bipartite join of Θ(3, 3) and K2 where Θ(3, 3)
is two vertices connected by three internally disjoint paths of length 3). There is also the following conjecture. For a family of graphs L, define ex(n, L) as the maximum number of edges in an n-vertex graph which con-tains no subgraph from the family L.
Conjecture 11. (Erd˝os-Simonovits) For every rational α ∈ (0, 1), there exists a finite family L for which ex(n, L) = Θ(n1+α).
Some interesting families of bipartite graph L are minors and topological subdivisions of complete graphs. We will not address these topic in the course but interested readers can refer to Diestel’s graph theory book.
5. Erd˝os-Simonovits and Sidorenko’s conjecture
Erd˝os and Simonovits’s conjecture addresses an interesting question re-garding multiplicity of bipartite graphs.
Conjecture 12. (Erd˝os-Simonovits) For every bipartite graph H, there ex-ist constants γ ∈ (0, 1), c > 0, and n0 such that graph G on n ≥ n0 vertices
with m ≥ n2−γ edges contains at least cn|V (H)|p|E(H)| copies of H where p = m
(n 2)
is the edge density of G.
Note that the expected number of copies of H in a random graph G(n, p) is roughly |Aut(H)|1 n|V (H)|p|E(H)|. Therefore the conjecture above in some
sense asserts that random graphs contain (or, is close to containing) the minimum number of copies of H.
Sidorenko made a similar conjecture on multiplicity of homomorphisms of bipartite graphs. A homomorphism of a graph H to G is a map f : V (H) → V (G) for which {f (v), f (w)} is and edge whenever {v, w} is an edge. Note that the map need not be injective. For two graphs H and G, let Hom(H, G) be the set of all homomorphisms from H to G, and let tH(G) = |V (G)||Hom(H,G)||V (H)|
be homomorphism density. We may interpret tH(G) as the probability that
a uniform random map from H to G forms a homomorphism. A beautiful conjecture of Sidorenko asserts that
Conjecture 13. For all bipartite graphs H and all graphs G, tH(G) ≥ tK2(G)
|E(H)|
.
This is an amazingly elegant conjecture; it does not require G to be a large graph and there is no constant involved. The conjecture fails for all non-bipartite graphs. For example if H is K3, then we can take G = K2 for
which the left-hand-side is 0 and the right-hand-side is 18.
When H is a bipartite graph consider the case when G is the random graph G(n, p). Then by linearity of expectation, we have |Hom(H, G)| = (1+o(1))n|V (H)|p|E(H)|(the o(1) term is there to account for the non-injective copies). Therefore tH(G) = (1+o(1))p|E(H)|. For H = K2, it gives tK2(G) =
(1 + o(1))p. Thus the inequality above is ‘almost tight’ for random graphs. Sidorenko’s conjecture asserts that for every bipartite graph H, the number of homomorphisms from H to G over n vertex graphs of fixed density is asymptotically minimized when G is random-like (this can be made formal but we will not go into detail at this point). Note that the case when H is a path is already non-trivial.
Surprisingly, the conjecture turns out to be equivalent to Erd˝os and Si-monovits’s conjecture. One direction is easy.
Sidorenko implies Erd˝os-Simonovits. Given a bipartite graph H, take γ = |E(H)|1 . Then the number of non-injective homomorphisms from H to G is at most n|V (H)|−1. On the other hand by Sidorenko’s conjecture, the number of homomorphisms from H to G is at least n|V (H)|p|E(H)|. Since p > 2n−γ (approximately),
n|V (H)|p|E(H)|− n|V (H)|−1≥ 1 2n
|V (H)|
p|E(H)|. Therefore the number of copies of H in G is at least
1 2|Aut(H)|n
|V (H)|
p|E(H)|. Erd˝os-Simonovits implies Sidorenko.
Lemma 14. For all G and N , there exists a graph G0 on at least N vertices for which tH(G0) = tH(G) for all graphs H.
Proof. Let G0 be a blow-up of G obtained by replacing each vertex with N vertices. Let π : V (G0) → V (G) be the canonical ‘projection’ homomor-phism. Note that h : V (H) → V (G0) is a homomorphism if and only if π ◦ h : V (H) → V (G) is a homomorphism. Therefore |Hom(H, G0)| = N|V (H)||Hom(H, G)| and tH(G0) = |Hom(H, G0)| |V (G0)||V (H)| = |Hom(H, G0)| N|V (H)||V (G)||V (H)| = |Hom(H, G)| |V (G)||V (H)| = tH(G). Suppose that Erd˝os-Simonovits conjecture is true but Sidorenko’s conjec-ture is not true. Let (H, G) be a pair that disproves Sidorenko’s conjecconjec-ture. Thus H is a bipartite graph and G is a n-vertex graph with p = tK2(G) and
tH(G) = (1 − ε)tK2(G)
|E(H)| where ε > 0. The following lemma is known as
the tensor power trick.
Lemma 15. For all G and k, there exists a graph G0 for which tH(G0) =
tH(G)k for all graphs H.
Proof. Let G⊗G be a graph with vertex set V (G)×V (G) where two vertices (v, w) and (v0, w0) are adjacent if and only if {v, v0} and {w, w0} are both
edges in G. We claim that graphs H,
Hom(H, G ⊗ G) ∼= Hom(H, G) × Hom(H, G)
where the correspondence is given by h ∈ Hom(H, G ⊗ G) mapping to π1◦ h
and π2◦ h for the projection maps π1 and π2 from V (G) × V (G) to the first
coordinate, and second coordinate, respectively. Therefore
tH(G × G) =
|Hom(H, G)|2
|V (G)|2|V (H)| = tH(G) 2.
The product G ⊗ G is known as the tensor product of G and G. One can similarly define tensor product for different graphs. The conclusion follows by considering the tensor product of G with itself k times. Let c be the constant coming from Erd˝os-Simonovits conjecture for H and define k as the minimum integer for which (1 − ε)k < c. By Lemma ?? there exists a graph G0 for which tH(G) = (1 − ε)ktK2(G)
|E(H)|< ct K2(G)
|E(H)|=
cp|E(H)|. On the other hand by Lemma ?? we may assume that the number of vertices of G0 is sufficiently large in terms of the edge density of G0. We must then have |Hom(H, G0)| ≥ c(n0)|V (H)|p|E(H)|. This is a contradiction since |Hom(H, G0)| = tH(G0) · (n0)|V (H)|.
We essentially proved that Sidorenko’s conjecture is equivalent to the following conjecture.
Conjecture 16. Let H be a bipartite graph. There exists a positive constant c such that for every p ∈ (0, 1), every large enough graph G with n vertices and at least 12n2p edges contains at least cn|V (H)|p|E(H)| copies of H.
Sidorenko’s conjecture is known to hold for few families of graphs: paths, trees, even cycles, complete graphs, hypercubes, etc. We say that a bipartite graph H has Sidorenko’s property if the conjecture holds for H.
5.1. Paths. Consider a path H = Pk where k ≥ 2 (path with k edges).
We prove that |Hom(H, G)| ≥ 2−2k−1nk+1pk holds for all n-vertex graphs with at least 12n2p edges. We will prove the statement by induction on n. If n = 1, then p = 0 and the claim trivially holds. Let G be an n-vertex graph with at least 12n2p edges and suppose that the claim has been proved for all smaller values of n.
Suppose that G has αn vertices of degree less than 14np. Let G0 be the graph obtained by removing all vertices of degree less than 14np. Note that G0 has (1 − α)n > 0 vertices and at least 12n2p − αn ·14np = 12(1 − α/2)n2p edges. Therefore if α > 0, then by the inductive hypothesis,
|Hom(H, G)| ≥ |Hom(H, G0)| ≥ 1 22k+1· (1 − α) k+1nk+1· 1 − α/2 (1 − α)2p k = 1 22k+1 (1 − α/2)k (1 − α)k−1n k+1pk ≥ 1 22k+1n k+1pk.
where the final inequality follows since k ≥ 2.
Hence we may assume that α = 0, i.e., that G has minimum degree at least 14np. In this case we can construct a homomorphism from Pk+1 to
G by arbitrarily choosing its first vertex and greedily extending it k times. This way each homomorphism is counted twice and therefore the number of paths of length k is at least
1 2n 1 4np k ≥ 1 22k+1n k+1pk.
Therefore Sidorenko’s conjecture holds for paths.
5.2. Complete bipartite graphs. Let H = Kt,s for some fixed integers t
and s. Let G be an n-vertex graph with m = 12n2p edges. The number of pairs (v, S) with |S| = s and S ⊆ N (v) is
M = X v∈V (G) deg(v) s ≥ nnp/2 s = (1 + o(1)) 1 2ss!n s+1ps
where the last inequality holds for large enough n depending on p (in other words K1,s satisfies Sidorenko’s conjecture). For a set S, let d(S) be the
number of common neighbors of S and note that M =P S⊆V (G),|S|=sd(S). Note that X S⊆V (G) d(S) t ≥n s ·M/ n s t ≈ n s s! · nps/2s t ≈ n t+s s!t!2tsp ts.
This proves that complete bipartite graphs satisfy Sidorenko’s conjecture. 5.3. Graphs with a ‘complete’ vertex. Conlon, Fox, and Sudakov proved that if H is a bipartite graph with two parts A∪B where there exists a vertex a0∈ A adjacent to all other vertices in B.
Let a = |A| and b = |B|. Let ε be a small enough positive real. For each integer k ∈ [b], we call a k-tuple of vertices X good if X has at least ε2npk common neighbors. Call it bad otherwise. Define Vk ⊆ V as the set
of vertices v for which N (v) contains at least ε deg(v)k bad k-tuples. Note that nk(ε2npk) ≥ εX v∈Vk deg(v)k≥ ε|Vk| 1 |Vk| X v∈Vk deg(v) k . Since |Vk| ≤ n, the above implies that
X
v∈Vk
deg(v) ≤ ε1/kn2p.
Define V0= V1∪ · · · ∪ Vb. Let U be the set of vertices having degree at least
εnp. Note that if ε is small enough, then X v∈U \V0 deg(v) ≥ 1 2n 2p − X v∈V \U εnp − X v∈V0 deg(v) ≥ 1 4n 2p.
Take a vertex v ∈ U \ V0. We say that a b-tuple of vertices X in N (v)
is excellent if every subset Y ⊂ X is good. By definition, the number of excellent b-tuples in N (v) is at least
deg(v)b− b X k=1 ε deg(v)k· deg(v)b−k· b k ≥ 1 2deg(v) b.
Consider a homomorphism from H to G where we map a0 ∈ A to v and
map B to some excellent b-tuple. The number of ways to embed a0 and B
as above is at least X v∈U \V0 1 2deg(v) b ≥ 1 2n 1 n X v∈U \V0 deg(v) b ≥ 1 2n · 1 4bn bpb = 1 22b+1n b+1pb.
For each a ∈ A other than a0, note that the image of N (a) is some good
deg(a)-tuple and hence has at least ε2npdeg(a) common neighbors. We may choose the image of a to be any one of such vertices. Therefore the number of
ways to extend the partial homomorphism above to a ‘full’ homomorphism is at least
Y
a∈A\{a0}
ε2npdeg(a) ≥ ε2(a−1)na−1pe(H)−b. Hence the total number of homomorphisms of H to G is at least
1 22b+1n
b+1pb· ε2(a−1)na−1pe(H)−b ≥ ε2(a−1)
22b+1 n
a+bpe(H).
This proves that Sidorenko’s conjecture holds for H.
This approach has been further generalized by Szegedy-Li, Kim-Lee-Lee, and Szegedy. The smallest graph for which Sidorenko’s conjecture is not known to be true is K5,5\ C10.
References
[1] M. Aigner, Tur´an’s Graph Theorem [2] N. Alon, J. Spencer, Probabilistic Method.
[3] D. Conlon, Extremal graph theory lecture notes (Lecture 1). [4] R. Honsberger, Mathematical Diamonds