Math 584:   Linear Algebraic Methods in Combinatorics; graduate course at UIUC.


Generic Syllabus

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.