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It is possible to avoid large homogeneous substructures in general networks, but a conjecture of Erdős and Hajnal from 1989 says that forbidding any specific substructure results in existence of a very large homogeneous one! In this article you will have a look into one of the most fascinating problems in modern graph theory.

If you see six people when you enter a class, it might be difficult to guess their personalities, but there is an amazing thing you can be sure of (mathematically prove): there are three of them who either all know each other, or who have never met. On the contrary, if there are only five people, there is a chance that you might fail to find such a triple: consider five people sitting around a circular table in a way that everybody knows only their two neighbors. Whoever three people you choose there will always be two people who know each other and two people who don't know each other, never all three.

Figure 1: Five people sitting around a circular table.

Why does this argument fail when you have six people sitting around a circular table? Think about this for a moment.

This simple observation due to Frank Ramsey in 1930 was the first step in mathematicians building a great theory on it ever since -- nowadays called Ramsey theory. It mainly concerns the structures emerging as the group gets larger. People in a class who may (or may not) have met before can be represented as a network of points that may (or may not) be linked to each other by an edge. There is a whole branch of mathematics that represents mathematical problems about networks in that way, called graph theory, and Ramsey theory is a subfield of graph theory.

Let us call a group of people homogeneous if either all know each other, or who have never met. Suppose instead of a group of three, you want to find a homogeneous group of size four. You can easily convince yourself that if, for instance, there are 10000 people in the class, this should be an easy task. But, what about the minimum size of the class in order to guarantee existence of such four people? The following example in the figure below shows you might fail if there are only 17 people:

Figure 2: An example of a graph on 17 vertices with no homogeneous set of size four

Indeed, it is the best example! It can be shown that one can always find a homogeneous group of size four in every group of 18 people.

Ready to be surprised? Mathematicians have no answer (yet) for the next obvious question:

Question: What is the minimum number of people to guarantee the existence of a homogeneous group of size five?

The question above is just one example of problems in graph theory which are innocent-looking but enormously difficult! Mathematicians are only able to establish that the answer should be a number between 43 and 48. How can this problem be not fully answered? I can hear you saying a computer would do the job, but it turns out that we need more than $10^{300}$ computations, which is quite beyond today's computer's abilities.

Let us start formalizing the concepts. In graph theory language, homogeneous groups correspond to substructures in graphs which are either all connected or all disconnected, respectively called cliques and independent sets. Letting $R(k)$ be the minimum integer $m$ such that every graph on $m$ vertices has a homogeneous substructure of size $k$, the previous discussion is mathematically equivalent to saying that $R(3)=6$ and $R(4)=18$. Recall that what we could only say about $R(5)$ was that $43\leq R(5)\leq 48$, so it seems there is no hope for $R(k)$ when $k\geq 6$, desperate! But, mathematicians never give up.

If something is quite difficult to determine exactly, it is natural to try to understand what it looks like by giving some good upper and lower bounds. The term asymptotic behavior for a function $f(x)$ informally means the categorization of $f(x)$ when $x$ grows towards infinity in terms of well-understood functions such as $x^n$ or $n^x$. For the case of our problem, the behavior of $R(k)$ when $k$ gets larger, the famous mathematician Paul Erdős, as the founder of the probabilistic method, used a probabilistic argument to show that $R(k)\geq (\sqrt{2})^{k}$. For the upper bound, Ramsey himself proved that $R(k)\leq 4^{k}$ by using a mathematical method called induction, still a huge gap!

The exact asymptotics of $R(k)$ -- whether $\lim_{k\to \infty}R(k)^{1/k}$ exists or not -- is still one of the biggest open problems in Ramsey theory, but now we are moving to the part in which we discuss another aspect of homogeneous structures. You can and look at proofs of aforementioned upper and lower bounds at the great reference $$Proofs from the book, Chapter 45: Probability makes counting (sometimes) easy$"$. Also you can read Nicos Starreveld's article about the latest breakthrough about the upper bound for $R(k)$ and further discussion.

Homogeneous substructures are unavoidable, but how large can they be?

Let us interpret the upper and lower bounds for $R(k)$ from another perspective: $R(k)\leq 4^k$ shows that every graph on $n$ vertices has a homogeneous substructure of size $\log_4 n$. On the other hand, Erdős' result $R(k)\geq (\sqrt{2})^k=4^{k/4}$ says that in an $n$-vertex graph, you can avoid homogeneous substructures of size $4\log_4 n$. The main message from these two facts can be summarized as follows:

• In an arbitrary graph, logarithmic sized homogeneous substructures are unavoidable.
• There are graphs whose largest homogeneous substructure has logarithmic size.

Erdős and his collaborator Andras Hajnal asked a question in a paper published in 1989 that is still quite open: Take any substructure you want, say $H$. If a graph does not have a part that exactly looks like $H$ in terms of edges and non-edges, it is called $H$-free. They conjectured that any $H$-free graph has to have a polynomial size homogeneous substructure -- rather than only logarithmic size you would see in an arbitrary graph (note that for any fixed positive numbers $\alpha$ and $\beta$, $n^\alpha$ becomes much larger than $\beta\log n$ when $n$ gets larger.).

Erdős-Hajnal conjecture
For every graph $H$, there exists a constant $\alpha\in (0,1)$ such that every
$H-$free graph $G$ on $n$ vertices has a homogeneous substructure of size $n^{\alpha}$.

Shall we believe in this conjecture?

At the first glance, it might not be intuitive why forbidding a substructure can drastically change the size of the largest homogeneous substructure in a graph compared to arbitrary graphs of the same size, but, of course, there are (mathematical) reasons for it. Maybe it is best to think about it first through an example. Let $H$ be a graph that contains three vertices and two edges, i.e. a substructure looks like a path of two edges, depicted in Figure 3 below.

Figure 3: A path graph $H$ with two edges.

Can we describe $H$-free graphs? Let $G$ be a $H$-free graph. Now we will try to understand how this affects the structure of $G$. Consider a vertex $v$ in $G$ which has at least two other neighbors (vertices adjacent to $v$ through a link). Call them as ${u_1,\ldots,u_k}$, depicted in Figure 4 below.

Figure 4: The vertex $u$ and its neighbors.

If $u_i$ and $u_j$ are not linked for some $i\neq j$, then observe that $u_i- v- u_j$ would be the same graph with $H$ (see Figure 5), a contradiction.

Figure 5: The vertex $u_i$ and $u_j$ are not connected, then $G$ contains $H$.

Therefore, all the neighbors of $v$ should be connected to each other, and this is true for any vertex with at least two neighbors. If you think about it for a few more seconds, you can conclude that $G$ should be a disjoint union of parts such that each part is a clique -- all vertices should be linked to each other within the substructure (treat a single vertex as a clique as well). Suppose $G$ consists of $k$ many cliques. Observe that the largest part has at least $n/k$ vertices, so we find a homogeneous structure of size $n/k$. On the other hand, we can choose one vertex from each part, they form again a homogeneous substructure because they are all disconnected. As a result, we can find a homogeneous substructure of size $\max\{k,n/k\}$. Since $k\cdot n/k =n$, either $k\geq n^{1/2}$ or $n/k\geq n^{1/2}$, so $\max\{k,n/k\}\geq n^{1/2}$. So, we proved the conjecture when $H$ is path graph consisting of two edges! One case is done among infinitely many graphs. You can try to prove one more case yourself.

Exercise: Let $H$ be the triangle graph, as depicted in Figure 6. Prove that every $H$-free graph on $n$ vertices has a homogeneous set of size $\sqrt{n}$.

Figure 6: A triangle graph $H$ consisting of three vertices which are all connected.

Solution: Let $G$ be a triangle-free graph on $n$ vertices. We will prove that $G$ has an independent set of size $\sqrt{n}$.

• Suppose there exists a vertex of at least $\sqrt{n}$ neighbors. Then, any two such neighbors …

• Suppose all the vertices have less than $\sqrt{n}$ neighbors. Consider the largest independent set $S$, then …

You can see that the conjecture is so strong because it is about any graph $H$. Desperately, since Erdős and Hajnal, there has been no significant progress with the exception examining some special cases of the graph $H$, just like we did. However, even the case of $H$ being a path of four edges (see Figure 7) is still unknown!

Figure 7: A path graph $H$ with four edges.

You can say that even if we could solve this case, it would not mean a huge step towards the general solution of the conjecture (because there are infinitely many graphs). You would be right, but as mathematician we do what we can do!

You can look at the papers listed at the end of this article to have an idea about the recent progress towards the conjecture, which makes mathematicians more optimistic about a proof in the near future. But who knows, maybe we still need to wait for a long time...

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