Problems

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.

Given r random vertices v_1,...,v_r of C^d , what is the expected number of 0 / 1-vectors in the affine subspace spanned by these vectors?

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)…

If D is a dimatroid and C,D ∈ D , then there exist C',D'∈ D of almost equal size whose union is C ∪ D .

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

In view of the absence of the Shannon effect and Kozik’s result on the limiting distributions it seems reasonable that asymptotically almost all functions have polynomial complexity.

Let ε > 0 be any constant and let q be a sufficiently large prime power. Let L be a set of at least q^5/2+ε lines in F_q^3 such that no plane contains more than (1/2)q^3/2 lines of L. Then, |P(L)| ≥ (1-o(1))q^3.

Is it true that f_DR^2(n,C)≈3/2n ? If not, can one prove at least, that f_DR^2(n,C)<(2-c)n? (34)

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 ?

For each vertex v in a graph G of order n ≥ 4, b(G-v)=b(G)+lfloorn/2⌋-2 if and only if G=C_4,P_4,2P_2

Find other possible values of the parameter d and the corresponding d-antimagic labeling of type (1, 1, 1) for the hexagonal plane map H_n^m.

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

Working in the differential tower of groups A imathS with A abelian of order r, when k ≤n the critical group K(V(U^kD^k)_n)=K(Ind_A imathS_n-k^A imathS_n1) is given, as a list of elementary divisors,…

In the general case, for any m > 1, n = 2^m, for any n-tuple of A matrices satisfying (1), does there always exist an n-tuple of B matrices of order c that satisfies construction (H0) under condition (H1), where c = M(n-1), with M defined…

For 0<α≤ 1 , among all trees, characterize the tree which has the maximum generalized distance spectral radius.

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

Let X,barψ be as in Lemma 6.2, and suppose that the roots of ∑_k=0^νp_kx^k are real. Then σ_ψ^2σ_X^2≥μ^4ν^-2 .

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^*).

Is the Merino–Welsh conjecture true for binary matroids?

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