Problems

For positive integers n_1 ≥ n_2 ≥ … ≥ n_t ≥ 2, we have R(P_2n_1, P_2n_2, …, P_2n_t) = n_1 + ∑_i=1^t (n_i - 1) + 1.

At the same time we suppose that the factor 3 in this condition is "conventional", i.e. we conjecture that for any natural k ≥ 3 after replacing this condition by p(r) ≥ kp Theorems 22-24 will remain true.

Open problem: • d ∈ 3, 4 for n ≥ 1

Let H be a hypertree. Then X_H is F-positive.

Conjecture 5.1. For β>0 , the only 2-ary, order 2(α,β) -Conolly recurrences are

The Möbius function of KC_n alternates in sign for each n ≥ 1.

Let χ(H)≤ k≥ 3 . Then rex(n,H,K_k+1)=(1+o(1))N(H,T(n,k)) .Moreover, if n is sufficiently large and is divisible by k, then rex(n,H,K_k+1)=N(H,T(n,k)) .

If given two positive integers a and b where a ≤ b ≤ 2a, is it possible to find a graph and a permutation α on V(G) such that γ(G)=a and γ(P_α(G))=b for all a,b ∈ N?

Find a derangement σ of the set M of all noncrossing perfect matchings of 1, 2, …, 2n such that ⟨ α, σ(α) ⟩ = 1 for all α ∈ M.

Let G be a finite graph and G = G(min). What condition on the set \mathcal{K}(G) = {K(v) | v \in V(G)} is equivalent to the existence of an endomorphism of G' modulo G?

Is M(u) a CFG or not?

Is this true? “For each positive integer n, there is a graph G with n vertices such that dis[G] - dis_s[G] = Ω(√n).”

Let φ be a primitive binary morphism with an antipalinidromic fixed point u. Then φ or φ^2 is conjugated to a morphism in class A_1 ∪ A_2.

This leaves open the question about the cardinality of the largest set of pairwise divergent infinite permutations.

α(k, r) = (k - 1)(4r - k - 6) for 5 ≤ r ≤ k ≤ 2r - 4.

Consequently one can ask if there is always an optimal circular permutation of the same form as the pattern.

Let P be a finite poset and ω a labeling of P. Then the following two conditions are equivalent. (i) ω is an admissible labeling. (ii) There exists m ∈ N such that ∑_φ ∈ A(P, ω) q^|φ| = q^m ∑_φ ∈ A(P) q^|φ|.

Let G be a tight graph such that: For every edge (u,v)∈ E(G) , one of its endpoints is dense, and the other is non-dense, and |N(u)∩ N(v)|≤ 1 , for all pair of vertices u,v ∈ D(G),u≠v .Then, χ_b(G) = m(G).

In particular, what is the dimension of the restriction of that space to patterns of size k? Is it spanned by corner trees with k vertices?

and for r ≥3 ∑_i=0^n(-1)^n-i( ln i )U_m+k+2+i,k+i^(r)= ll2^n if m=2n, 2^n(5/2n+2r-1) if m=2n+1, . (1.11)