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Is C_4 in DP? More generally, are even cycles in DP?

Let w ∈ S_n be a permutation and l := ℓ(w) be its length. Denote by CS(w) = a = (a_1 ≤ a_2 ≤ … ≤ a_l) ∈ N^l the set of compatible sequences [7] corresponding to permutation w. Define statistics r(a) on the set of all compatible sequences…

Is it possible to partition K_9^3 into stars S_4 so that their mates partition K_9^4? (Star partition without the mate condition is possible [5].)

Can we prove that they are actually convergent to the same limit?

The magnitude homology of a graph obtained by gluing two cycle graphs C_3 along single edges to a single cycle graph C_4 has diagonal magnitude homology provided those triangles are not attached to opposite sides of the 4-cycle.

Does lim_n → ∞ hatr_∞(G_n) = 0 hold for every sequence of graphs (G_n) such that |V(G_n)| → ∞ and Δ(G_n) is bounded as n → ∞? What sequences (G_n) yield lim_n → ∞ hatr_∞(G_n) = 1?

Are there some graphs with diam(G)=n and diam(D_{2}(G))=⌈(1/2)diam(G)⌉+1.

Let f(x_1,…,x_n) be a polynomial over a field F given by (1.1) and (1.2). Provided n≥ k, for any finite subset A of F we have |(f(x_1,…,x_n):x_1,…,x_n∈ A, and x_i≠ x_j if i≠ j)| ≥ minp(F)-llbracket n=2 a_1=-a_2rrbracket,…

The maximum possible load, on any vertex in any graph, is 1/8n^3-O(n^2) .

It is an interesting open problem to classify all fixed points of the twist map, and to determine whether V_k,n is the only totally positive fixed point.

For any d and any connected simple graph G of order d, is i(P_G,m) always a stable polynomial?

We believe that equality should hold in Theorem 1.

Given a finite CW complex X which is not contractible and two ρ-immersed matroids (M, l) and (N, l') such that T_X(M, l) ≃ T_X(N, l'), if there exists a surjective weak map τ: M → N, then τ^# is an isomorphism.

But we conjecture that at least one of π and π^{-1} will always have a sufficiently large strong compatible set to ensure a better approximation for bs(π)=bs(π^{-1}) .

The radius of G_n is equal to n-σ(n)-1 .

A minimal forbidden induced subgraph for the property c_2(G)≤ k has at most 2 k+2 vertices.

Conjecture 6.2. Fix k,n \in N . If \leq is standard and x_{k-1}x_{l+1}=x_{k}x_{l} for all 0<k \leq l, then the sequence (e_{k-j}(n+j))_{j\geq 0} is PF with respect to \leq .

For every graph G = (V + s, E) with d(s) ≥ 4 there exist rs, st ∈ E such that for every best-balanced orientation vecG_rt of G_rt, vecG := vecG_rt - rt + rs + st is a best-balanced orientation of G.

Given a non-negative number σ, is there a graph G with diameter 2, degree Δ and Δ^2+1-σ vertices?

We discuss ... including some new ones that we present in this last section (in particular Conjectures 11.3 and 11.5).