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For any r ≥ 2 there exists n_0 = n_0(r) such that if F ⊆ 2^[n] is r-wise intersecting with n ≥ n_0, then the number of (r+1)-triangles in F is at most N(Δ_r+1, F_X), where F_X = F ⊆ [n] : |F ∩ X| ≥ |X|-1 for some (r+1)-set X.

Let G be a class 1 regular graph with Δ > n/3. If any graph obtained from G by splitting a vertex is a critical class 2 graph.

Is the double sequence A^1,id vertically C-log-concave?

Find other possible values of the parameter d and the corresponding d-antimagic labeling of type (1, 1, 1) for the generalized Petersen graph P(n, 2).

Characterize the (m, n)-extremal graphs for all m and n.

It is open if this is true for all cycle-free graphs E, which would show that the assumption "S idempotent" can be dropped.

We believe the assertion of Theorem 1.6 is true even for all smaller values of n, though we don't have a proof yet.

Determine whether μ_s(D_n)=n+1for even integers n with n \ge 6.

Do subcubic graphs have exponentially independent sets of linear order?

(3) χ(G(2,9,7))=4?

While we are not able to prove it, we think it is a very safe conjecture that R_2(m) converges to below 0.607 , as m tends toward infinity.

Based on the values for |X_n| for small values of n we conjecture that |X_n| = o(n), and leave open the question of enumerating the members of X_n in ascending order, in O(|X_n|) time.

Are there any other sequences (a_n)_n ∈N of integers appearing 'naturally'with the property that there exists a real α>0 such that (α a_nbmod 2 π)_n=1^∞ has an absolutely continuous non-uniform distribution?

It would be interesting to determine whether there are schemes for which the fusing-relations graph and fusing-idempotents graph are not isomorphic.

SM_{P_{h}}(P_{n})=\frac{4nh-3n+h-(lh+th-l^{2}-t^{2}-t)}{2} for any integer 2 \le h \le n and 1 \le l \le h, 1 \le t \le h-1 such that n \equiv l (\bmod h) and n-1 \equiv t (\bmod (h-1)).

Can occurrences of mesh patterns be used to compute the Betti numbers of permutation complexes? Or can we define a set of mesh patterns P such that if π avoids P its permutation complex is contractible?

We do not succeed to solve the case D which is left as an open question (for this case, values in Table 1 are experimentally obtained).

Does there exist a finite t_0 such that d_t(H)=d_t_0(H) for t≥t_0?

We conjecture that this property is not valid if m>4.

Is there an integer n_0 , where n_0<2(k+r) , such that for any n≥n_0 , we have γ_k(n,r)≥γ_k(n+1,r)?