Mathematics / advanced
Extremal and structural graph theory
Find forced patterns and sharp bounds using density, regularity, expansion, Ramsey theory, and graph limits.
Your route through mathematics
Follow the path or begin with a question. Every lesson is open.
Your place, at your pace. Progress is optional and saved only in this browser when available.
Read / explore / reason
Inside this module
Module 01 advanced
Extremal and structural graph theory
Find forced patterns and sharp bounds using density, regularity, expansion, Ramsey theory, and graph limits.
- 01
Turán's theorem and extremal density
Find the sharp edge threshold forcing a clique, including the balanced multipartite equality case and exact finite-size conventions.
29 min - 02
Supersaturation and counting subgraphs
Turn a fixed density excess into polynomially many copies and track labeled versus unlabeled subgraph counts.
30 min - 03
Ramsey numbers and recursive bounds
Prove finite two-color clique guarantees, construct a sharp small example, and derive a probabilistic lower-bound criterion.
30 min - 04
Extremal bipartite graphs and Zarankiewicz bounds
Count common neighborhoods to bound forbidden complete bipartite subgraphs, and explain why bipartite extremal problems need finer scales.
30 min - 05
Spectral graph bounds and expansion
Connect eigenvalues to edge discrepancy and bottlenecks while separating connectivity, bipartite periodicity, and mixing assumptions.
33 min - 06
Szemerédi's regularity lemma
Define uniform density across large subsets, state a precise equitable partition theorem, and understand the bounded-energy refinement mechanism.
34 min - 07
Counting lemmas and graph removal
Turn regular dense pairs into many embeddings and convert a sparse pattern count into a small deletion certificate.
34 min - 08
Graph minors and treewidth
Measure tree-like structure with bags, distinguish minors from subgraphs, and state the grid-minor obstruction with its proper quantifiers.
33 min - 09
Extremal hypergraphs and codegrees
Extend edge and degree counting to uniform set systems and diagnose why high local codegree can coexist with a global matching obstruction.
34 min - 10
Dense graph limits and graphons
Represent dense graph sequences by measurable kernels while separating homomorphism, induced, and finite-edge density conventions.
35 min