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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).

Let p ≥ 2 be an integer and G be a nonbipartite graph of order n, with minimum degree δ>2n/(2p+3) . Then G contains a cycle of length l, for each integer l,2p≤l≤δ+1 .

Conjecture 1.6.1. The polynomial f_m(b,q) has the form f_m(b,q)=∑_i=0^binomm2(1-q)^m-y^(i)g_m,i(q)b^i where y(n)=⌊frac√8n+12⌋ and g_m,i(q) are polynomials. Further, with <_k^n> denot ing the Eulerian numbers ^3…

There is an isomorphism of graded S_n-modules R_n,k,r≅ R_n,k,r' .

A natural conjecture would be that for any (0,1)-matrix, the lattice formed by its integral null vectors has a small number of near-shortest vectors.

The case σ = 2 can be described using hypergeometric functions; is there a notion of generalized hypergeometric function that could be applied for larger values of σ?

Determine the compound curling numbers different products of graphs in which one graph is a regular graph.

Interestingly, all the properties of Proposition 2 hold even for negative k, and it seems that for any k the A_k(n) eventually become positive for n sufficiently large, ...

(4) χ(G(2,11,9))=4?

In particular, is it true that if the realization |Γ| of Γ through its direct complex Δ(Γ) is a manifold, then the realization of its partial dual |Γ^S| is also a manifold?

Every graph without isolated vertices admits super edge total local antimagic labeling.

For central arrangements whose underlying matroid is connected, the homotopy type of the complement determines the underlying matroid.

Every triangularly connected P_3 -dominated graph on at least three vertices is vertex pancyclic, with an exception K_1,1,3 .

Let 1 ≤ t ≤ r ≤ binomn2. If n is sufficiently large relative to t and r, then the set B_r(n) is t-EKR.