It is very hard to determines(C) for an arbitrary closure C. However, there are nice combinatorial results for certain closures.
Definition 4.2.1 Let Cnk denote the following closure on R:
Cnk(X) =
X if |X|< k
R otherwise. (4.7)
The following lemma gives a general lower bound fors(Cnk).
Lemma 4.2.2 ([DK81])
s(Cnk) 2
≥ n
k−1
. (4.8)
Proof of Lemma 4.2.2 Suppose, thatM representsCnk and let|A|=k−1 be a subset ofR, furthermore letb6∈A. Then by the definition ofCnk,A6→b, i.e., there is a pair of rows i and j, such that they are identical in A, but different in b. If there is another k−1-subset B of R such that i and j are identical on B, as well, then A∪B 6→ b would hold, but |A∪B| ≥ k, so by the definition of Cnk this cannot happen. Thus, we can assign distinct pairs
of rows to distinctk−1-subsets of columns.
The exact value ofs(Cnk) is determined for certain values ofk.
Theorem 4.2.3 ([DK81]) The following equalities hold: We give the proof of Case b) as an example.
Proof of Case b) of Theorem 4.2.3 Let s =s(Cn2). Lemma 4.2.2 gives
s 2
≥ n. Note that the number of the right hand side of equality in Case b) is the smallest s satisfying the previous inequality. If s is such, then we construct a matrix M with s rows such that CM =Cn2 as follows:
There is a pair of zeros in every column ofM such that for different columns the zeros are in different pairs of rows, which implies that every one-element subset of R is closed. This can be done by the choice of s. On the other hand, no two rows agree in more than one column, so ifA⊆Rwith|A|>1,
then CM(A) =R.
Let us note that in Case d) of Theorem 4.2.3 Lemma 4.2.2 yields onlys(Cnn)>
√2n, hence some other tricks are needed to prove n+ 1 as a lower bound.
Let us now consider the case k= 3. From Lemma 4.2.2 we obtain that s(Cn3)
Consider the dual problem. A column naturally determines a partition of the set Y of rows, by the equalities of its entries. We say that a partition covers the pair (α, β) (α, β ∈ Y, α6=β) iff α and β are in the same class of the partition. We can state the previous two properties as follows.
Findn partitions of Y (|Y|=n) such that:
1’) for any two partitions there exists a pair (α, β) covered by both, 2’) no pair (α, β) is covered by three different partitions.
However, the number of pairs of partitions is also n2
and different pairs of partitions cannot cover the same pair of elements by 2’). Thus, we may conclude that 1’) and 2’) (consequently 1) and 2)) are equivalent to:
(i) for any two partitions there is exactly one pair of elements, which is covered by both,
(ii) each pair of elements is covered by exactly two different partitions.
Definition 4.2.4 A collection of partitions satisfying (i) and (ii) is called an orthogonal double cover.
The following conjecture was formulated in [DFK85]. (It was posed in other terms, since the notion of orthogonal double cover was introduced later, in [GGM94].)
Conjecture 4.2.5 There exists an orthogonal double cover of the n-element set by n partitions provided n ≥7.
In the same paper they proved that Conjecture 4.2.5 is true for certain n’s.
Theorem 4.2.6 ([DFK85]) Conjecture 4.2.5 is true if n = 12r+ 1or n = 12r+ 4.
In the proof they used a theorem of Hanani to construct special type of partitions, namely each partition consisted of one 1-element class and 4r (4r+ 1, resp.) 3-element classes. This motivated the following conjecture.
Conjecture 4.2.7 ([DFK85]) Ifn= 3r+1, then there exists an orthogonal double cover of then-element set bynpartitions that have one 1-element class and r of the 3-element classes.
Note that the two conjectures are independent in the sense that the solution of one of them does not imply the solution of the other. The first result about these conjectures was negative. In 1987 Rausche [Rau87] observed that Conjecture 4.2.7 is not true forn = 10. However, that turned out to be the only ”bad case”. Ganter and Gronau and later Yeow Meng Chee [Che92]
independently proved the following.
Theorem 4.2.8 ([GG87]) Conjecture 4.2.7 is true for n≥13.
The first conjecture was decided affirmatively, as well. Bennett and Wu proved the following theorem.
Theorem 4.2.9 ([BW90]) Conjecture 4.2.5 is true.
If we have an orthogonal double cover by partitions, then we can define a graph for each partition. The vertex set is the underlying set R, the edges are the pairs covered by that partition. These graphs are unions of disjoint cliques. Furthermore, in the case of Conjecture 4.2.7 these graphs are pairwise isomorphic, namely they are unions of r K3’s and an isolated point. This observation motivated the following definition.
Definition 4.2.10 A collection of n pairwise isomorphic graphs with the same vertex set V, where |V| = n and Gi = (V, Ei) for i = 1,2, . . . , n, is called an orthogonal double cover by graphs iff
1) each edge of Kn is contained in exactly two of the Ei’s, 2) |Ei∩Ej|= 1 for i6=j.
With this concept Theorem 4.2.9 states that there exists a double cover by graphs where the Gi = K1 +r ∗K3. Gronau, Mullin and Schellenberg [GMS95] proved a conjecture of Chung and West [CD94] stating that there is an orthogonal double cover by graphs where eachGi has maximum degree at most two. A sharpening of this result was given by Ganter, Gronau and Mullin.
Theorem 4.2.11 ([GGM94]) For all n ≥4, n 6= 8 there is an orthogonal double cover by graphs where each Gi consists of the isolated vertex i and a union of disjoint cycles of length 3,4 or 5 only.
An extensive survey of orthogonal double covers is found in [Gro02]
The exact value of s(Cnk) is not known for k > 3. However, if k is fixed, then its asymptotic behavior is known.
Theorem 4.2.12 ([DFK85]) If k is fixed and n > n0(k), then
c1(k)nk−12 ≤s(Cnk)≤c2(k)nk−12 . (4.12) The lower bound in Theorem 4.2.12 follows from Lemma 4.2.2. The upper bound is proven by a construction involving polynomials over a finite field.
F¨uredi proved some bounds for the ”other end” of the range of k.
Theorem 4.2.13 ([F¨ur90]) If k is fixed and n > n0(k), then
c3(k)n2k+13 ≤s(Cnn−k)≤c4(k)nk. (4.13) The following concept allows us to find s(C) for infinitely many closures.
Definition 4.2.14 Let L and N be closures on the ground sets U and V, respectively, with U ∩V =∅. The direct product of L and N is the closure on the ground set U ∪V defined by
(L × N)(A) = L(A∩U)∪ N(A∩V) for A⊆U ∪V. (4.14) The size of a minimum representation of a direct product of closures can be calculated provided that the minimum representation known for the members of the product.
Theorem 4.2.15 ([DFK85])
s(C1× C2) =s(C1) +s(C2)−1. (4.15) Theorem 4.2.15 provides an alternative proof for Case d) of Theorem 4.2.3, one has only to observe that Cnn =Cn−1n−1 × C11.