Math 584: Linear Algebraic Methods in Combinatorics; graduate course at UIUC.
Potential Homework problems. Any comment or new problems are welcome.
Lectures: M stands for the Matousek book. Note that many lectures are taking more than one class. I used 42 fifty minutes classes to cover all. The notes are mostly self-contained, one can skip some or change the order of reading them.
Lecture I: M1: Fibonacci numbers, M3: Oddtown, M4: Eventown, M5: Generalized Fisher inequality, M6: Odd distances, M7: Euclidean distances.
Lecture II: M8: Decomposition of cliques into complete bipartite graphs, M9: Equiangular lines, M10: Finding triangle, M11: checking matric multiplication, M12: Tiling square with rectangles.
Lecture III: M13: Decomposing K_10 into Petersen graphs, M14: Hoffman-Singleton M15: Two-distances between points.
Lecture IV: M16: covering hypercube - 0, M17: Avoiding medium size intersection is hard, M18: Reducing the diameter of a set.
Lecture V: M19: Beck-Fiala Theorem, M20: Balancing vectors, M21: Matrix-tree theorem (number of spanning trees), Bollobás Coffee Time #75: Coloring of K_n with specific K_4s.
Lecture VI: M22: Number of perfect matchings in bipartite graphs, M23: Unimodularity of number of partitions.
Lecture VII: M24: Perfect matchings and Determinants; M25: Kakeya Needle Problem (by Dvir).
Lecture VIII: Bucic-Letzter-Sudakov-Tran: Minimum saturated families of sets.
Lecture IX: M28: Shannon Capacity; M29: Vertex disjoint union of graphs might have large Shannon Capacity (by Alon).
Lecture X: Friendship Theorem (Erdõs-Rényi-Sós), Bootstrap percolation on the square grid, The rectangle process on the grid, Bollobás (skew) set-pair inequality with linear algebraic proof, application to weak clique saturation.
Lecture XI: M33: Exterior Algebras. Proving Bollobás skew-set-pair inequality.
Lecture XII: Bootstrap percolation on the hypercube, Morrison-Noel Theorem, Hambardzumyan, Hatami, Qian approach.
Lecture XIII : Conlon-Ferber, Widgerson, Sawin: Multicolor-Ramsey.
Lecture XIV : Combinatorial Null-StellenSatz.
Lecture XV: Applications of Null-StellenSatz: Berge-Sauer-Tashnikov, Alon-Friedland-Kalai, Hypercube-0 covering by hyperplanes, Permanent Lemma, Erdõs-Ginzburg-Ziv, Alon-Tarsi.
Lecture XVI: Huang Hao: Sensitivity Conjecture.
Lecture XVII : Eigenvalues of graphs, basic properties, interlacing theorems, Bussemaker theorem.
Lecture XVIII: Alon-Chung expander mixing Lemma, applications to expanders, proving Erdõs-Ko-Rado theorem.
Lecture XIX: Finite geometry, Reimann C_4-free construction, Sidon sets upper bound, Erdõs-Turán construction, Ruzsa construction, C_4-free polarity graphs.
Lecture XX: K_3,3-free graph construction (no proof), Conlon: C_4, C_6, C_10-free constructions.
Lecture XXI: Ferber: Singularity of symmetric random matrix.
Lecture XXII: Behrend construction, Random algebraic constructions K_s,t-free (sketch).
Lecture XXIII: Katz-Kropp-Maggioni: K_2,2,2-free hypergraphs (sketch).
Lecture XXIV : Babai-Frankl book: Nagy Construction for Ramsey(t,t), Deza-Frankl-Shingi, Alon-Babai-Suzuki, Ray-Chaudhary-Wilson.
Lecture XXV: Balogh-Linz: Frankl-Tokushige, Prague dimsension, Bollobás skew set-pair, proved by linear algebra, clique saturation.
Lecture XXVI: Bollobás Coffee Time problems #41, #141, #142: Equivalent versions of Borsuk Theorem, Hobby- Rice Theorem, Cutting necklace, Alon-West Theorem.
Lecture XXVII: Progressions-free sets in Z_4^n are exponentially small; by Croot-Lev-Pach.
Lecture XXVIII: M30: Equilateral sets: Alon-Pudlák.
Lecture XXIX: 3-AP-free subsets of F_q^n are exponentially small by Ellenberg-Gijswijt.
Lecture XXX: Rédei-Rényi difference bases (Bollobás Coffee Time Problems); Discrepancy of 2-colorings of AP-s, by Roth; Introduction to basic Fourier Methods.
Note that some of the lectures notes are covering more than one 50 minutes class.