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If vecG is a best-balanced orientation of G := (V + s, E) and varrho_vecG(s) = δ_vecG(s) then there exist rs, st ∈ A(vecG) so that vecG_rt is a best-balanced orientation of G_rt.

For any positive integers k and n satisfying k < n, and any alternating function f: [k] × [k] → Z_n, there exists a permutation π ∈ S_k such that d_π(i, j) ≠ f(i, j) pmodn, for all distinct i, j ∈ [k].

We don't know if the theorem 2 is true whenever p=ω , m=ω_1,n=ω_2 and n^p=ω_3=2^p : we do not suppose g.c.h. .

A regular, k-intersecting hypergraph on n vertices has at most 2^{n-(2^{k+1}-k-2)} edges when k \ge 3.

Let f be a finite field. Suppose barX, barY, barα, and barZ_f are random variables with values in f; barα is distributed with respect to the probability counting measure on the set f^× of non-zero elements of f, and barZ_f is distributed…

Let G be a bipartite graph with sides A and B, and each edge colored red or blue. For a set X ⊆ A let N^RB(X) denote the set of vertices, which are joined to X by a red and by a blue edge as well. Suppose |N^RB(X)| ≥ |X| - 1 holds for…

The natural conjecture is that the Hasse index of any order h ≥ 1 is asymptotically Boolean, i.e. that i_h(D_n)=fracsc_h(D_n)|D_n|fracn^h2^h (for n →+∞ ) for every h ≥ 1.

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 .