Problems

For every r ≥0 and every h≥h0(r), where h0(r) depends on r, there is an integer p1(h,r) depending on h and r, such that for every p≥p1(h,r), each member of K^{-r}(p,p+h) is χ-unique.

Let K_n_1,n_2,...,n_r be a complete r-partite graph with r vertex sets X_i(i ∈[1,r]) and |X_i|=n_i , ∑_i=1^r=n . Besides (i) and (ii) in Theorem 8, fgndi_∑(K_n_1,n_2,...,n_r)≤ 3 ?

Is this true “for any graph G, dis_s[G] ≤ χ(G)?”

For any nonnegative integer k, does there exist a connected graph G satisfying φ_(2,j)(G) − κ_(2,j)(G) + 1 = k?

Is there a graph G such that for any non-empty graph H, we have χ_{td}(H □ G) > χ_{td}(G)?

For any x ∈(-3,c) with x ≠ 0, we have ∑_k=0^∞fracx^3k(k+1)(x-1)^k( c3k k )= frac3(x-1)2x^3log^2(1-x)+frac2(1-x)x^3q(x)^2 +s(x)log(1-x)+fract(x)q(x)√(1-x)(3+x), (1.17) where s(x) and t(x) are suitable rational functions in x.

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.