Combinatorics and graph theory II.
9th practice, 24th of November, 2021.
Sperner, LYM, Erd˝ os-de Bruijn
Good to know
Erd˝os-Ko-Rado theorem. (1961) IfF ⊆2[n] is ak-uniform set-system (k < n/2) with the property that for anyA, B∈ F :A∩B6=∅, then|F | ≤ n−1k−1
and this can be attained.
De Bruijn – Erd˝os theorem(1948)F ⊂2[n], for allA∈ F |A| ≥2, and for arbitrary elements 1≤i < j≤n there is exactly oneA∈ F such thati, j∈A. Then|F |= 1 or|F | ≥n.
Sperner’s theorem (1928)F ⊂ 2[n], for allA, B ∈ F: A 6⊂ B and B 6⊂ A. Then |F | ≤ bn/2cn
. In case of equality:F= bn/2c[n]
or F= dn/2e[n]
.
LYM inequality F ⊂2[n], for all A, B ∈ F:A 6⊂B and B 6⊂A. Letfk denote the number of k element sets contained inF. ThenPn
k=0 fk
(nk) ≤1. In case of equality:F= [n]k
for somek.
1. LetT be annvertex tree.
(a) At most how manyconnected subgraphsofT can be choosen if none of them is a subgraph of another one?
(b) At most how manyinduced subgraphs ofT can be choosen if none of them is a subgraph of the another one?
2. Show a set system F ⊆ 2[n], such that the intersection of any two sets contained inF contains at least two elements and |F |= 2n−2. Is there a bigger F ⊆2[n] which satisfies the first proeprty? Useful:: 2kk
≤22k−1. (Furthermore,≤22k/√
k ifk is big enough.)
3. Let F be a set system which does not contain a chain of size s+ 1 (so there are no sets A1, A2, . . . As+1 in F such that A1 ⊂ A2 ⊂ · · · ⊂ As+1). Prove that Pn
k=0 fk
(nk) ≤s, where fk is the number of k element sets contained inF..
4. Let F ⊆ 2[2n] be a set system such that the cardinality of each set contained in F is even and any two set contained inF intersect each other. Show that ifnis even, then F contains at most 22n−2 sets.
5. LetF ⊆2[n] be a set system such that the cardinality of each set contained inF is even but the cardinality of the intersection of any two sets contained inF is odd. Show that |F | ≤ bn/2cn
.
6. Assume that any two edges of the hypergraph H = (V,E) are either disjoint or one of them contains the another. At most how many edges can Hhave if it does not have repeated edges?
7. n people are living in a village. How many clubs can be established in the village such that ifAi denotes the members of clubi, then|Ai| 6≡0 (mod 3) and for alli6=j:|Ai∩Aj| ≡0 (mod 3). (*)
8. Assume thatk < n/3 andF ⊆ [n]k
is ak-uniform set-system which does not contain 3 pairwise disjoint sets.
Show that|F | ≤4 n−1k−1 .
9. Assume, that the hypergraphH= (V,E) does not contain a cycle, so there is no sequence containing pairwise disjoint verties and hyperedges such thatx1, E1, x2, E2, . . . xk, Ek, xk+1=x1 whereEi contains verticesxi and xi+1. Show that if the∅ is not an edge ofHand His connected (V is not a disjoint union of two nonempty sets V1 andV2such that any hyperedge is a subset ofV1 orV2), thenP{|E| −1 :E∈ E}=|V| −1.
Homework
1. Assume that the set system F ⊆ 2[n] does not contain two sets which are disjoint. Show that there is a set systemF0 ⊆2[n] which contains F as a subsystem such that F0 does not contain two sets which are disjoint and
|F0|= 2n−1.
2. For eachk≥1 show ak-uniform hypergraph which is isomorphich to its dual hypergraph.
