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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,…

Computations in some cases with n = 3 suggest that if S is a generating set for Γ0, then F1 = F0 and ⟨S⟩Γ is precisely the collection of points L of Γ for which there exists an i-element E of Δ(F0) such that L ⊂ E^⟂ (in Δ(F)) and L ∩ E ≠ ∅.

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.

Let C be a projective [n,k]_q three-weight code with non-zero weights w_1 < w_2 < w_3 satisfying w_1 + w_2 + w_3 = 3(1-1/q)n. Then w_2 = (1-1/q)n. Moreover, w_1 = w_2 - t and w_3 = w_2 + t, where t is a power of the characteristic p of F_q.

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…

There exists a constant c>1, such that for any triangle-free planar n-vertex graph G,P_DP(G,4)≥ c^n .

We conjecture a hexagon conditional local move as well:

Problem 6.1. Suppose 1/n ≪ε ≪ 1/k . Let X, Y be disjoint sets with |X|=ε n and |Y|=n. Let c=c(n) and t=t(n) with c≤ t ≤|X| . Let H be a k-graph on X ∪ Y with δ_k-1(H)≥ t-c such that H[Y] is independent. What is the complexity of deciding…

Fix d ≥ 2 and ζ > 0. Suppose G is an k-uniform set system on [n], where ζ n < k < n/2, n is sufficiently large, and either G contains no strong d-simplex or G contains no d-cluster. If |G| > (1+ζ)binomn-2k-2, then G is a star.

If k is a power of a prime, P_L_k(G,k)=P_DP(G,k).