Problems

Do subcubic graphs have exponentially independent sets of linear order?

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?

Let G=(V, E) be a locally finite, recurrent graph which is quasi-isometric to R. Let h be a harmonic function on G, and suppose that for some finite cut E(X, Y) separating the two ends of G we have ∂h(X,Y)=0. Then h is either constant or…

Consider maximal planar graphs, 3-connected planar graphs or planar 3-trees. For all such graphs G, is there a constant α<1 and a constant k such that Z(G)≤ α n+k ?

CONJECTURE M(D) : For all A ∈∂ Ω(R_n),f_A(λ)=per(λ A+(1-λ)D_n) is nondecreasing in the interval 0 leqqλ leqq 1 .

I.e., for hereditary property H, what are the maximal properties H' ⊇ H such that d^*(H') = d^*(H)?

Do trees of maximum degree at most 4 have exponentially independent sets of linear order?

If the linear compression dimension of S is given by log_{2}(|S|)+o(n) (hence matches the non-linear compression dimension) then S is contained in the union of 2^{o(n)} translates of some subspaces of size at most |S|.

Let k ≥5 be an odd integer and G be a (n,d,λ) -graph satisfying d^k-1≫ λ^k-2 . Then G has global resilience (1 / 4+o(1)) n d with respect to being C_k -free.

An interesting question is whether every red-blue coloring of a k-pseudorandom graph contains a monochromatic path of length \Omega(\frac{n}{\sqrt{k}}).

It is not known whether there is a finite k-chromatic graph of girth at least g and with \chi_{c}^{s}(G)=2k.

We conjecture that uniqueness of the MCB implies U ≠ ∅.

Does there exist a graph Γ of order n and level of symmetry equal to n-k/n for arbitrarily large k ≥2 ?