组合数学 (Spring 2014)/Problem Set 4

Problem 1

An [math]\displaystyle{ n }[/math]-player tournament (竞赛图) [math]\displaystyle{ T([n],E) }[/math] is said to be transitive, if there exists a permutation [math]\displaystyle{ \pi }[/math] of [math]\displaystyle{ [n] }[/math] such that [math]\displaystyle{ \pi_i\lt \pi_j }[/math] for every [math]\displaystyle{ (i,j)\in E }[/math].

Show that for any [math]\displaystyle{ k\ge 3 }[/math], there exists an [math]\displaystyle{ N(k) }[/math] such that every tournament of [math]\displaystyle{ n\ge N(k) }[/math] players contains a transitive sub-tournament of [math]\displaystyle{ k }[/math] players.

Problem 2

Let [math]\displaystyle{ G(U,V,E) }[/math] be a bipartite graph. Let [math]\displaystyle{ \delta_U }[/math] be the minimum degree of vertices in [math]\displaystyle{ U }[/math], and [math]\displaystyle{ \Delta_V }[/math] be the maximum degree of vertices in [math]\displaystyle{ V }[/math].

Show that if [math]\displaystyle{ \delta_U\ge \Delta_V }[/math], then there must be a matching in [math]\displaystyle{ G }[/math] such that all vertices in [math]\displaystyle{ U }[/math] are matched.

Problem 3

Prove the following statement:

• For any [math]\displaystyle{ n }[/math] distinct finite sets [math]\displaystyle{ S_1,S_2,\ldots,S_n }[/math], there always is a collection [math]\displaystyle{ \mathcal{F}\subseteq \{S_1,S_2,\ldots,S_n\} }[/math] such that [math]\displaystyle{ |\mathcal{F}|\ge \lfloor\sqrt{n}\rfloor }[/math] and for any different [math]\displaystyle{ A,B,C\in\mathcal{F} }[/math] we have [math]\displaystyle{ A\cup B\neq C }[/math].

(Hint: use Dilworth theorem.)

Problem 4

A Boolean matrix is a matrix whose entries are either 0 or 1. We call a matrix 1-matrix if all its entries are 1.

Let [math]\displaystyle{ A }[/math] be a Boolean matrix satisfying:

• there are totally [math]\displaystyle{ m }[/math] 1-entries in [math]\displaystyle{ A }[/math];
• every row of [math]\displaystyle{ A }[/math] contains at most [math]\displaystyle{ s }[/math] 1-entries;
• [math]\displaystyle{ A }[/math] does not contain any [math]\displaystyle{ r\times r }[/math] 1-matrix as submatrix.

Use König-Egerváry Theorem to prove: we need at least [math]\displaystyle{ \frac{m}{sr} }[/math] many 1-matrices to precisely cover all the 1-entries in [math]\displaystyle{ A }[/math].