An open workspace for mathematical discovery. Check proofs, make comments, form collaborations.

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)?

We conjecture that the actual number is in fact exponential in n^l.

For any integer n ≥ 6, we have β_b(K_1 ⊙ P_n) = ⌈ n/6 ⌉ + 1

For which sequences n_i, k_i, does the comb graph C(k_i, n_i) have Ulam number 2?

Every (anti)palindromic graph has order multiple of four (plus two).

β_2(k, r) = 4r - k - 2 for 2 ≤ r < k < 2r - 1.

Let G and H be two bipartite graphs. If G ~ H, prove (or disprove) that N_K(m,n)(G)=N_K(m,n)(H) for all m, n such that 2 ≤ m ≤ n.

We leave open, if this hold for property F in general.