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Let (G, p) be a generic framework in R^d. If (G, p) is globally (d, k)-rigid and G is not complete, then there exists σ ∈ ker DR_k(G, p)^T such that rank Ω(σ) = |V| − d + k − 1.

Is there some C(r)>2 such that ρ_2(2(r+2)+1,r+2)≥C(r)ρ_2(2r+1,r) holds for all r ≥2 ?

For n<30 the maximal coefficient is not uniquely attained only for n=2,5,6,12,13,14and 15. Are those the only cases when this happens? If not, can we predict when?

Finally, we conjecture that, for every set A of integers, deciding whether a digraph has a handle decomposition with all handles of length in A is NP-complete, unless there exists h ∈ N such that A = 1, …, h.

Let n ≥k and t ≤ c = n/k. Let P ⊂ U_k^n be a partially t-intersecting partition system. Then, |P| ≤ (\begin{array}{c}n-tc-t\k-1\end{array}) U(n-c,k-1). Moreover, this bound is tight if and only if P is equal (up to permutations of [1, n])…

Is C_τ(A_+) always isomorphic to C_π(A_+) ? In other words, is there a bijection f:C_τ→ C_π that satisfies the condition d(x,y)∈ A_+⇔ d(f(x),f(y))∈ A_+?

Let \lambda be a partition and \mu , and \eta be compositions such that |\lambda|=|\mu| and ll(\mu)\le |\eta| . Does there exist a quiver Q, dimensional vector \beta and GL(Q,\beta) -weight \sigma such that…

What is, for a given integer k ≥ 1 and any C (if k = 1, then C ≥ 1), the minimum m(C) such that any graph G with mad(G) ≤ 2k - m(C) satisfies χ_l(G^2) ≤ kΔ(G) + C.

It remains an open question as to whether ζ^*(n, S_7) grows faster than cubically.

Let Γ be a connected t-valenced graph with two main and two plain eigenvalues. There exists a positive integer C such that if t ≥C, then Γ is a strong graph.

Suppose G is a Hamiltonian chordal graph. Is G cycle extendable if min{Δ(T) : (T, T) is a tree decomposition for G} = 4?

In an intersecting r-partite hypergraph, what is the smallest size of a vertex cover that does not contain any edge or side?

Let b_t(n) denote the minimum number of edges induced by any set of n / 2 vertices in the Turán graph on n vertices for K_t .If each set of ⌊ n/2 ⌋ vertices in a graph G of order n spans more than b_t(n) edges, then G contains a K_t .

Conjecture 4.2. Let h≥1,(a_{1},...,a_{h})\in \mathbb{C}^{h} , and P(x)\in \mathbb{C}[x] . Set I_h,P,n(x)=P(x)∏_i=1^n(1+a_1x^F_i+a_2x^F_i+1+… +a_hx^F_i+h-1).Regarding h, P as fixed, let c_{n}(p) denote the coefficient of x^{p} in…

There exists a constant K such that for every integer m, where m ≥ K, there exists an integer n such that χ_st'(C_m square C_n) = 6.

Let Y and Z be t-cross-intersecting sets in G_n whose sizes meet the bound in Theorem 1.2. If n is sufficiently large compared to t, then Y = Z and Y or Y^T is a t-coset.

For an arbitrary planar graph G, is there a proper grid drawing of G in a grid of polynomial size?

Is it true that for sufficiently large n, indeed we can have min_i ∈[n]deg_B(i)deg_C(i)≤binomn-2k-2^2?

Does there exist a Hall function for the pairwise disjointness relation of the sets of a given family F?

It seems an interesting problem to characterize semisymmetric graphs with Wiener dimension 2.