Modern examples of probabilistic method in combinatorics

Classic graph-theoretic examples of probabilistic method in combinatorics

If you want to understand the spirit of the method, the best examples live in graph theory. The core template is always the same:

1. Define a random object (graph, set system, coloring).
2. Show that the probability it has the desired property is greater than zero.
3. Conclude that some object with that property must exist, even if you never write it down explicitly.

Let’s walk through several classic examples of examples of probabilistic method in combinatorics that every serious combinatorics student should know.

Example 1: Lower bounds for Ramsey numbers

Ramsey theory asks: how large does a graph need to be before it must contain a large clique or a large independent set? The diagonal Ramsey number \(R(k,k)\) is the smallest \(n\) such that every red–blue coloring of the edges of the complete graph on \(n\) vertices contains either a red \(K_k\) or a blue \(K_k\).

A famous example of the probabilistic method shows that
[
R(k,k) > 2^{k/2}
]
for large \(k\). Here’s the sketch of the argument.

We color each edge of \(K_n\) independently red or blue with probability \(1/2\). For a fixed set of \(k\) vertices, the probability that all edges among them are red is \(2^{-\binom{k}{2}}\); the same for all blue. By a union bound, the probability that this particular \(k\)-set is monochromatic is at most \(2 \cdot 2^{-\binom{k}{2}}\).

There are \(\binom{n}{k}\) choices of \(k\)-sets, so the expected number of monochromatic \(K_k\) subgraphs is [ \mathbb{E}[X] \le \binom{n}{k} \cdot 2^{1-\binom{k}{2}}. ] If we can choose \(n\) so that this expectation is less than 1, then there **exists** a coloring with no monochromatic \(K_k\), hence \(R(k,k) > n\). A simple estimate shows this works for (n \approx 2^{k/2}.

This is one of the best examples of how probabilistic reasoning beats deterministic construction: we still do not know explicit colorings that match these bounds, but the probabilistic method guarantees they exist.

For a detailed, rigorous treatment, the classic reference is Noga Alon and Joel Spencer’s The Probabilistic Method (3rd ed., Wiley), which is widely used in graduate courses and research seminars.

Example 2: Graphs with large girth and large chromatic number

Another striking example of examples of probabilistic method in combinatorics is due to Paul Erdős. Intuitively, a graph with large girth (no short cycles) feels “tree-like,” and trees are 2-colorable. So you might expect that forbidding short cycles makes coloring easier.

Erdős used the probabilistic method to prove the opposite: for any integers \(g\) and \(k\), there exists a finite graph with

• girth at least \(g\) (no cycles shorter than \(g\)), and
• chromatic number at least \(k\).

The proof starts from a random graph \(G(n,p)\). One shows that, with positive probability, the graph has few short cycles and yet has no large independent set. Then you delete one vertex from each short cycle. The resulting graph still has high chromatic number (because no large independent set suddenly appears) and now has large girth.

This is a textbook example of the probabilistic method: it proves the existence of extremely counterintuitive objects without constructing them explicitly.

For background on random graphs and this style of argument, see lecture notes on \(G(n,p)\) models from MIT OpenCourseWare (e.g., 18.404 Theory of Computation, or 18.226 Random Graphs and Networks) hosted at mit.edu.

Set systems and extremal examples of probabilistic method in combinatorics

The probabilistic method is just as powerful for set systems and extremal combinatorics. These examples include some of the cleanest applications of expectation and variance.

Example 3: Large subsets with no 3-term arithmetic progression

A classic problem: how large can a subset of \({1,2,\dots,n}\) be if it contains no 3-term arithmetic progression (3-AP), i.e., no triple \(a, a+d, a+2d\)?

One probabilistic argument (often used as a warm-up to Roth’s theorem and later results) goes like this. Choose each integer from \(1\) to \(n\) independently with probability \(p\). Let \(X\) be the size of the random subset; then \(\mathbb{E}[X] = pn\).

Let \(Y\) be the number of 3-term arithmetic progressions contained in that subset. One can bound \(\mathbb{E}[Y]\) by counting all 3-APs in \({1,\dots,n}\) and noting that each survives with probability \(p^3\). With a careful choice of \(p\) (for instance, \(p\) on the order of \(n^{-1/2}\) in a simple version), you can show
[
\mathbb{E}[X - Y] > 0.
]

Now, delete one element from each 3-AP in your random set. The resulting set has no 3-APs and still has size at least \(X - Y\). Since the expected value of \(X - Y\) is positive, there exists a 3-AP-free subset of size on the order of \(n^{1/2}\).

Modern results (like the Green–Tao theorem on primes in arithmetic progressions) use much deeper tools, but this remains one of the best examples for teaching the basic probabilistic method.

For an accessible introduction to additive combinatorics, including probabilistic flavors, see Terence Tao’s lecture notes hosted at math.ucla.edu.

Example 4: The probabilistic method and the Erdős–Ko–Rado style problems

In extremal set theory, a common theme is: given a family of subsets of \({1,\dots,n}\) with some intersection property, how large can the family be?

Probabilistic arguments often provide sharp or near-sharp upper bounds by considering a random object (like a random permutation) and looking at how many sets in the family satisfy some event. For instance, in variations of the Erdős–Ko–Rado theorem, one picks a random permutation of \({1,\dots,n}\) and studies the probability that a given set appears in a particular position. By linearity of expectation, you get constraints on the total number of sets in the family.

While many core theorems in this area have combinatorial proofs, probabilistic proofs are frequently shorter and more flexible, which is why these are often cited as clean examples of examples of probabilistic method in combinatorics.

Concentration, random graphs, and modern real examples

The “first generation” of examples relied mostly on linearity of expectation and Markov’s inequality. Modern combinatorics leans heavily on concentration inequalities (Chernoff bounds, Hoeffding’s inequality, Azuma–Hoeffding martingale bounds) and random graph models.

These tools show up in real examples across theoretical computer science and network science.

Example 5: Thresholds in random graphs \(G(n,p)\)

Consider the Erdős–Rényi random graph \(G(n,p)\), where each edge appears independently with probability \(p\). One of the most influential examples of probabilistic method in combinatorics is the study of threshold functions: sharp values of \(p\) where a property suddenly becomes likely.

Some well-known thresholds:

• Connectivity appears around \(p \approx (\log n)/n\).
• The appearance of a Hamiltonian cycle has a similar threshold.
• The giant component emerges when \(p \approx 1/n\).

The proofs use probabilistic estimates to show that, for \(p\) below the threshold, the property fails with high probability, and for \(p\) above, it holds with high probability. While this is not the traditional “existence by positive probability” template, the philosophy is the same: randomness is the tool to understand what structures must or must not exist at a given edge density.

These results are now standard in advanced courses on random graphs and probabilistic combinatorics; see, for example, notes from courses at Princeton University or Harvard University that cover \(G(n,p)\) and phase transitions.

Example 6: The Lovász Local Lemma and rare bad events

Sometimes simple union bounds are too weak. You have many “bad” events, each with small probability, but their union still looks too large. The Lovász Local Lemma (LLL) is one of the best examples of probabilistic method in combinatorics for handling this situation.

A classic application: proper colorings of hypergraphs, or avoiding small monochromatic structures in colorings of large sets. The LLL says, roughly, that if each bad event has small probability and is only dependent on a limited number of other events, then with positive probability none of them occur.

Real examples include:

• Constructing colorings of hypergraphs with no monochromatic edge.
• Building sparse set systems that avoid forbidden configurations.
• Designing error-correcting codes with certain local constraints.

For instance, consider a \(k\)-uniform hypergraph where each edge has \(k\) vertices. You randomly color the vertices with a fixed number of colors. A bad event is: “this particular edge is monochromatic.” Each event is rare, and each edge only shares vertices with a bounded number of other edges. Under the LLL conditions, there exists a coloring with no monochromatic edge.

The algorithmic version of the LLL (Moser–Tardos) has become central in theoretical computer science, giving constructive versions of previously non-constructive examples of probabilistic method in combinatorics.

Coding theory and theoretical computer science: more real examples

Probabilistic constructions are now standard tools in theoretical computer science, especially in coding theory, pseudorandomness, and complexity theory. These are very much real examples with 2024–2025 relevance.

Example 7: Existence of good error-correcting codes

Error-correcting codes are used everywhere from deep-space communication to consumer electronics. A central question: for given block length \(n\) and rate \(R\), how large can the minimum distance be?

The Gilbert–Varshamov bound is a classic example of probabilistic method in combinatorics applied to coding theory. Consider all binary strings of length \(n\). Repeatedly pick a random string and delete all strings within a certain Hamming distance from it. Probabilistic counting shows that you can continue this process long enough to build a large code with guaranteed minimum distance.

The argument is inherently probabilistic: you show that a random choice of codewords, subject to simple constraints, almost surely yields a code close to optimal. This is why many of the best examples in modern coding theory start with a random construction, even if later work finds explicit, algebraic codes with similar parameters.

For a rigorous coding theory introduction, see course materials from institutions like MIT or Stanford, which often include probabilistic constructions.

Example 8: Expander graphs and randomness extractors

Expander graphs are highly connected sparse graphs with strong mixing properties. They are vital in derandomization, network design, and cryptography.

One classic probabilistic argument: a random \(d\)-regular graph on \(n\) vertices is an expander with high probability for suitable \(d\). This gives a clean example of probabilistic method in combinatorics: it shows that “most” regular graphs are excellent expanders, long before we had explicit constructions of comparable quality.

These ideas feed directly into randomness extractors and pseudorandom generators, which are active research areas in 2024–2025. Many state-of-the-art constructions still start from or are analyzed via probabilistic methods, even when the final objects are explicit.

Why these are the best examples to study first

If you’re building intuition, the examples above form a natural progression:

• Start with expectations and union bounds (Ramsey lower bounds, 3-AP-free sets).
• Move to more structured random objects (random graphs, random set systems).
• Add concentration inequalities and the Lovász Local Lemma.
• Then look at applications in coding theory, expanders, and theoretical computer science.

These examples include both classical theorems and modern applications, and together they give a realistic picture of how working combinatorialists actually use probability in 2024–2025.

For a deeper dive, besides the Alon–Spencer text, you might look at graduate course notes in probabilistic combinatorics hosted at major universities (.edu domains) and survey articles from organizations like the American Mathematical Society (ams.org). Those sources frequently present new, real examples of probabilistic method in combinatorics that update and extend the classic toolkit.

FAQ: examples of probabilistic method in combinatorics

Q1. What is a simple example of the probabilistic method a beginner can understand?
A very approachable example of probabilistic method is the proof that there exists a graph on \(n\) vertices with more than half of the maximum possible number of edges and no triangles. You take a random graph \(G(n,p)\) with a carefully chosen \(p\), compute the expected number of edges and the expected number of triangles, and then delete one edge from each triangle. The expectation argument shows that some graph must remain with many edges and no triangles.

Q2. Which examples of probabilistic method are most important for contest math (like the Putnam)?
For contest settings, the best examples to know are: (1) Ramsey-type lower bound arguments using random colorings, (2) expectation-based constructions of large subsets avoiding a forbidden pattern (like 3-term arithmetic progressions), and (3) simple applications of Markov’s and Chebyshev’s inequalities to show that a random object almost surely has some property.

Q3. Are there real examples of probabilistic method used outside pure math?
Yes. Network science, information theory, and theoretical computer science are packed with real examples. Random graphs model social and communication networks; probabilistic existence proofs underlie bounds in coding theory and cryptography; and randomized constructions of expanders and hash families are standard in algorithm design.

Q4. Where can I read more detailed, rigorous proofs of these examples of probabilistic method in combinatorics?
The go-to reference is Alon and Spencer’s The Probabilistic Method. For free online material, graduate-level course notes from universities like MIT, Harvard, and others on probabilistic combinatorics or random graphs are excellent. Many of these are publicly available on .edu domains and walk through these examples in full detail.