组合数学 (Spring 2026)/Problem Set 2

Problem 1

Suppose [math]\displaystyle{ n \geq 4 }[/math], and let [math]\displaystyle{ H }[/math] be an [math]\displaystyle{ n }[/math]-uniform hypergraph with at most [math]\displaystyle{ \frac{4^{n−1}}{3^n} }[/math] (hyper)edges. Prove that there is a coloring of the vertices of [math]\displaystyle{ H }[/math] by four colors so that in every (hyper)edge all four colors are represented.

Problem 2

Use the Lovász Local Lemma to show that, if [math]\displaystyle{ 4\binom{k}{2}\binom{n}{k-2}2^{1-\binom{k}{2}} \leq 1 }[/math], then it is possible to color the edges of [math]\displaystyle{ K_n }[/math] with two colors so that it has no monochromatic [math]\displaystyle{ K_k }[/math] subgraph.

Problem 3

Let [math]\displaystyle{ G = (V, E) }[/math] be an undirected graph and suppose each [math]\displaystyle{ v \in V }[/math] is associated with a set [math]\displaystyle{ S(v) }[/math] of at least [math]\displaystyle{ 2\mathrm{e}r }[/math] colors, where [math]\displaystyle{ r \geq 1 }[/math]. Suppose, in addition, that for each [math]\displaystyle{ v \in V }[/math] and [math]\displaystyle{ c \in S(v) }[/math] there are at most [math]\displaystyle{ r }[/math] neighbors [math]\displaystyle{ u }[/math] of [math]\displaystyle{ v }[/math] such that [math]\displaystyle{ c }[/math] lies in [math]\displaystyle{ S(u) }[/math]. Prove that there is a proper coloring of [math]\displaystyle{ G }[/math] assigning to each vertex [math]\displaystyle{ v }[/math] a color from its class [math]\displaystyle{ S(v) }[/math] such that, for any edge [math]\displaystyle{ (u, v) \in E }[/math], the colors assigned to [math]\displaystyle{ u }[/math] and [math]\displaystyle{ v }[/math] are different.

Problem 4

Solve the following two existence problems:

• You are given [math]\displaystyle{ n }[/math] integers [math]\displaystyle{ a_1,a_2,\dots,a_n }[/math], such that for each [math]\displaystyle{ 1\leq i\leq n }[/math] it holds that [math]\displaystyle{ i-n\leq a_i\leq i-1 }[/math]. Show that there exists a nonempty subsequence (not necessarily consecutive) of these integers, whose sum is equal to [math]\displaystyle{ 0 }[/math]. (Hint: Consider [math]\displaystyle{ b_i=a_i-i }[/math])

• You are given two multisets [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math], both with [math]\displaystyle{ n }[/math] integers from [math]\displaystyle{ 1 }[/math] to [math]\displaystyle{ n }[/math]. Show that there exist two nonempty subsets [math]\displaystyle{ A'\subseteq A }[/math] and [math]\displaystyle{ B'\subseteq B }[/math] with equal sum, i.e. [math]\displaystyle{ \sum\limits_{x\in A'}x=\sum\limits_{y\in B'}y }[/math] (Hint: Replace the term multiset by sequence, the term subset by consecutive subsequence, and the statement is still true. )

Problem 5

Let [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] be positive integers, and let [math]\displaystyle{ (A_1,B_1),(A_2,B_2),\ldots,(A_m,B_m) }[/math] be [math]\displaystyle{ m }[/math] ordered pairs of finite subsets of the positive integers. Suppose that for every [math]\displaystyle{ i=1,2,\ldots,m }[/math], [math]\displaystyle{ |A_i|=a, |B_i|=b, A_i\cap B_i=\varnothing. }[/math]

Prove that if [math]\displaystyle{ m\gt \binom{a+b}{a}, }[/math] then there exist two distinct indices [math]\displaystyle{ i\neq j }[/math] such that [math]\displaystyle{ A_i\cap B_j=\varnothing. }[/math]

Problem 6

For a positive integer [math]\displaystyle{ n }[/math], write [math]\displaystyle{ [n]=\{1,2,\ldots,n\}. }[/math]

A set [math]\displaystyle{ A\subseteq [n] }[/math] is called subset-sum-distinct if the [math]\displaystyle{ 2^{|A|} }[/math] sums [math]\displaystyle{ \sum_{a\in S}a, S\subseteq A, }[/math] are pairwise distinct, where the empty sum is understood to be [math]\displaystyle{ 0 }[/math].

Define [math]\displaystyle{ f(n) }[/math] to be the largest integer [math]\displaystyle{ k }[/math] for which there exists a subset-sum-distinct set [math]\displaystyle{ A\subseteq [n] }[/math] with [math]\displaystyle{ |A|=k }[/math].

Prove the following two upper bounds.

• Prove that there is an absolute constant [math]\displaystyle{ c }[/math] with the following property: [math]\displaystyle{ f(n)\leq \log_2 n+\log_2\log_2 n+c. }[/math]

• Prove that there is an absolute constant [math]\displaystyle{ c }[/math] with the following property: [math]\displaystyle{ f(n)\leq \log_2 n+\frac12\log_2\log_2 n+c. }[/math]