Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
This is the Final Exam of Mathematical Tripos which includes Lie Groups, Lie Algebras, and Their Representations, Homomorphism, Inner Derivations, Irreducible Root System, Real Vector Space, Weyl Group, Irreducible Summands, Space of Diagonal Matrices etc. Key important points are: Combinatorics, Kruskal-Katona Theorem, Ahlswede-Khachatrian Theorem, Uniform Hypergraph, Family of Pairs of Subsets, Sauer-Shelah Lemma, Collection of Finite Subsets, Hypergraph of Order
Typology: Exams
1 / 2
This page cannot be seen from the preview
Don't miss anything!
Tuesday, 1 June, 2010 9:00 am to 12:00 pm
Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
2
1
State and prove the Kruskal-Katona theorem.
Show that a graph with 15 edges may contain 15 copies of K 4 but no more.
Let Ai = {A โ n^ : |A โฉ [t + 2i]| > t + i} and nk = (r โ t + 1)(2 + (t โ 1)/k). By comparing Ai+1 \ Ai with Ai \ Ai+1 , or otherwise, show that if nk+1 < n < nk then maxi |Ai| = |Ak|.
State and prove the Ahlswede-Khachatrian theorem giving the value of M (n, r, t), the maximum size of a t-intersecting family A โ n.
[You may use lemmas about compressions preserving t-intersections, and the exis- tence of a generating family on a small ground-set, provided you state them clearly.]
What does it mean that an r-uniform hypergraph is strongly (r + t)-saturated?
( Show that a strongly (r^ +^ t)-saturated^ r-uniform hypergraph of order^ n^ has at least n r
(nโt r
edges.
Let (Ri, Si), i โ I be a family of pairs of subsets with Ri โ n^ and Si โ n, such that Ri โฉ Sj 6 = โ if and only if i = j. Show that if |I| > 2 then |I| 6 n โ r โ s + 2 and that equality can be attained.
[Hint: pick xi โ Ri โฉ Si .]
State and prove the Sauer-Shelah lemma on families that shatter k-sets.
Let A be a collection of finite subsets of some infinite set X and, for Y โ X, let A|Y = {Y โฉ A : A โ A}. Suppose that, for every k > 1, there is a set Y โ X(k)^ with |A|Y | > 2 kโ^1. Show that, for every k > 1 , there is a set Z โ X(k)^ that is shattered by A. Must there be an infinite Z โ X such that A|Z contains every finite subset of Z?
Part III, Paper 10
