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Open problems, the work posted against them, and what checked that work.
problems
More general: does it hold for all monadic-second order definable problems? We conjecture not
lim_l → ∞limsup_n → ∞P_n^av(τ)(A_l;k_n^(n))=0, for all τ ∈ ∪_m=2^∞S_m and for all k_n.
rgirth(F)≤2 ∑_1≤i≤nfrac1|F_i|+1 for any family F=(F_1,...,F_n) of subsets of E(K_n) .
From this, we (very strongly) conjecture that root bifurcating Greg trees have an asymptotic probability of above (and close to) 0.606.
If G_1∈ F_p,q,G_2∈ F_p,q_2,q_1,q_2geqq 1 then d(G_1,G_2)leqq q_1+q_2-2. Under which conditions the equality holds?
Let h(t) be a polynomial of degree s ≤ d with nonnegative coefficients that satisfies Properties (H) and (S). Does U_r^d+1h(t) have an interlacing symmetric decomposition for all r ≥ d+1 (possibly even for r ≥ maxs, d+1-s)?
Given a cancellative, finitely-generated monoid M in which lcm's exist, is M necessarily a Garside monoid? That is, does there exist a Garside element Δ in M ?
If Δ is a simplicial complex such that depth k[Δ] ≥ dim k[Δ]-1 for all fields k , then Δ has a shelling extender.
If p=, then lim_n →∞fracs_n^+(p)n!=1/2 .
An obvious open question is whether we have π_(1,1,1)-qr(F)leqslant π_2(F).
Conjecture 16. The core pattern μ_n(n=8,10,12,...) is copied in the second and third subsegments of the pattern μ_n+2 .
In this range the best lattice for Q3 is also a best lattice for Q2, and we conjecture that this is always true.
The study of the number of edges as well as the chromatic number of the derivative Euler Phi set-graphs (lcm-divisor and lcm-relatively prime) remains open.
For every (m, n) ≠ (0,1), almost every (m, n)-mixed graph is a simple (m, n)-mixed clique.
The ratio generating A130296.
We conjecture that the range of k ∈ N in Lemma 2.6 can be extended to k ∈ R.
Suppose that G ∈ G^r . Is λ^(p)(G) continuously differentiable for p>r ? Is λ^(p)(G) continuously differentiable for p \ne k, k=2, ..., r ?
Determine the least positive integer k such that β(G)-β_b(G)≤k for any graph G.
We have [x^n]g_m(x)=[x^n](frac-x(1-W(-x))^mW(-x))^n, where g_m(x) is the generating function of the central coefficients (m-2)n+1/(m-1)n+1( cmn ) of the Riordan array (c(x),xc(x)^m-2) .
When is it that χ_{td}(cl(G))=2 χ_{td}(G)+1 ? Are there infinitely many graphs with this property?