Problems

Maximal in size concept lattice of a formal context (G, M, I) of VC-dimension at most k, such that |G| + |M| = 2n, and such that k divides n, is the Cartesian product of k chains of length n/k - 1 each: L = bigotimes_k C(n/k), where C(l)…

What is limsup_n →∞frace(G)binomnr:G ⊆( c[n] r ),G containsnosubgraphthatcoverspairs?

Given k, does every circle in an edge-minimal k-highly connected standard subspace X of |G| contain a vertex or end whose degree in X is at most k ?

Are there families of graphs such that the independence equivalence class is unbounded and each independence polynomial is irreducible?

Every t-design of 2 t+k elements can be obtained from k points in t-good position using the methods developed here.

Let 3≤k≤ℓ and n≥2 ℓ . If G is an n-vertex k-chromatic ℓ -connected graph and t ≥ 3, then i_t(G)≤ i_t(G^*).

An interesting open problem is to find all the values that can be attained from paths.

We conjecture that a 3-edge-connected, nonplanar graph with representativity at least 5 has exponentially many peripheral cycles.

Open problem: • d ∈ 3, 4 for n ≥ 1

The triple (L_([2] × [2]) × [2](R^a+3), Pro, ξ(f, (x_1, x_2), a) + ξ(f, (3-x_1, 3-x_2), a)) is 4-mesic.

It seems reasonable to conjecture that for degree sequences of any order n ≥ 1 the modal multiplicity will be 1, while the median and mean multiplicities will both increase with n, the latter much more rapidly than the former.

We formulate an analogue of Conjecture 1.2 for term orders with x_1 > x_2 > … > x_n (Conjecture 11.15).

For an integer k ≥ 2 and a sufficiently large n. Let G be an n vertex C_3ℓ+1-free graph for every integer ℓ ≥ k. Then for every r, 3k ≥ r ≥ 2, the number of cliques of size r in G is at most n-1/3k-1 binom3kr. Equality holds only for…

If we assume this conjecture, then we show that 1/k!∑_c ∈ LQ(m,k)ε(c)≥ 0 . This is still an open problem.

We conjecture that any such A can be covered by O(1) homothetic copies of C(B) with total volume O(|A|).

Let G be an (n, λ)-connected graph and let S be a given subset of V(G) such that |S| = n. Then G has λ edge-disjoint cycles C_1, ..., C_λ such that S ⊆ V(C_i) for all i, 1 ≤ i ≤ λ.

K(n, m) \le \lceil (m-1)d^2/n + 1/m \rceil, where d = \lfloor n/m \rfloor.

The condition in Theorem 4 is necessary as well as sufficient.

We leave as Conjecture 5.38 that this also holds for n even.

What is the computational complexity of COMPLETE WIDTH on 2K_2 -free graphs?