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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.

Construct and analyze analogues of the Hanoi graph H_n in variants 4 and 5. Is there a generalization of the Lucas Correspondence that works here?

Let a_n=Δ_k(n) , then for n ≥ 14 and k=1 or 2 I(a_n,a_n+1,a_n+2,a_n+3,a_n+4) =A(a_n,a_n+1,a_n+2,a_n+3,a_n+4)^3 -27B(a_n,a_n+1,a_n+2,a_n+3,a_n+4)^2>0, where A(a_n,a_n+1,a_n+2,a_n+3,a_n+4)=a_na_n+4-4a_n+1a_n+3+3a_n+2^2,…

Can you show F(n; 2)< (1+ε)n/2 ? What about larger values of t ?

Does there exist an infinite family of connected graphs F such that, src(G) is bounded on F, while s r v c(G) is unbounded?

For every 2-coloring of the positive integers there exist positive integers a,d such that the elements of the arithmetic progression a,a+d,...,a+d^2 all are colored the same.

π(K_5^=) = 227/387 and σ(K_5^=) = 2909/8127.

Suppose m, n ∈ N, m is odd, n is even and n > m. Suppose also that C_m × C_n is not one of the cases considered in Theorem 5. Then ρ(C_m × C_n) = 3.

Given an arbitrary rational number q ∈ [0,1] does there exist a relevant triple ⟨n,k,⋄⟩ such that q = Ψ(n,k,⋄)/n^k?

If W(n,n,w)>0 for some n and w, then W(m,m,w)>0 whenever m≥n.

If both G and barG are connected, then γ_e(G)≤ 3d_c(barG).

For any ℓ ∈N and any (not necessarily finite) partition N=∪_i=1^∞C_i , there exists d ∈N such that either (i) for some i_0∈N we have C_i_0∩(C_i_0-d)∩(C_i_0-2d)∩… ∩(C_i_0-ℓ d)≠∅ , or (ii) for every i ∈N we have (C_i-jd)∩(C_i-kd)=∅ for all…

We conjecture that a version of this representation result holds even for convex geometries in which the empty set is not closed.

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…