For a graph H, the extremal number ex(n,H) is the maximum number of edges in a graph of order n not containing a subgraph isomorphic to H. Let δ(H) > 0 and ∆(H) denote the minimum degree and maximum degree of H, respectively. We prove that for all n sufficiently large, if H is any graph of order n with ∆(H) ≤ √ n/200, then ex(n,H) = ( n−1 2 ) +δ(H)−1. The condition on the maximum degree is tigh...

A certain inequality is shown to hold for the values of the Möbius function of the poset obtained by attaching a maximum element to a lower Eulerian Cohen– Macaulay poset. In two important special cases, this inequality provides partial results supporting Stanley’s nonnegativity conjecture for the toric h-vector of a lower Eulerian Cohen–Macaulay meet-semilattice and Adin’s nonnegativity conjec...

A graph G is called chromatic-choosable if χl(G) = χ(G). It is an interesting problem to find graphs that are chromatic-choosable. There are several famous conjectures that some classes of graphs are chromatic-choosable including the List Coloring Conjecture, which states that any line graph is chromatic-choosable. The square G of a graph G is the graph defined on V (G) such that two vertices u...

A well-known conjecture of Stanley is that the h-vector any matroid a pure O-sequence. There have been numerous papers with partial progress on this conjecture, but it still wide open. Positroids are special class linear matroids play crucial role in field total positivity. In short note, we prove Stanley’s holds for positroids.

Let F be a totally real number field and let p be a finite prime of F , such that p splits completely in the finite abelian extension H of F . Stark has proposed a conjecture stating the existence of a p-unit in H with absolute values at the places above p specified in terms of the values at zero of the partial zeta-functions associated to H/F . Gross proposed a refinement of Stark’s conjecture...

We consider an extremal problem for graphs as introduced by Erdős, Jacobson and Lehel in [7]. Let π be an n-element graphic sequence. Let H be a graph. We wish to determine the smallest m such that any nterm graphic sequence π whose terms sum to at least m has some realization containing H as a subgraph. Denote this value m by σ(H, n). For an arbitrarily chosen H, we give a construction that yi...

We introduce here a new universality conjecture for levels of random Hamiltonians, in the same spirit as the local REM conjecture made by S. Mertens and H. Bauke. We establish our conjecture for a wide class of Gaussian and non-Gaussian Hamiltonians, which include the p-spin models, the Sherrington-Kirkpatrick model and the number partitioning problem. We prove that our universality result is o...

Abstract A long-standing conjecture of Erd?s and Simonovits asserts that for every rational number $r\in (1,2)$ there exists a bipartite graph H such $\mathrm{ex}(n,H)=\Theta(n^r)$ . So far this is known to be true only rationals form $1+1/k$ $2-1/k$ , integers $k\geq 2$ In paper, we add new which the true: $2-2/(2k+1)$ This in turn also gives an affirmative answer question Pinchasi Sharir on c...

A conjecture of Gabbay (1981) states that any class of flows of time having the property known as finite H-dimension admits a finite set of expressively complete one-dimensional temporal connectives. Here we show that the class of ‘circular’ structures refutes the generalisation of this conjecture to Kripke frames. We then construct from this class, by a general method, a new class of irreflexi...

A conjecture for the projection norm (or Lebesgue constant) of a weighted interpolation method based on the zeros of Chebyshev polynomials of the third and fourth kinds is resolved. This conjecture was made in a paper by J. C. Mason and G. H. Elliott in 1995. The proof of the conjecture is achieved by relating the projection norm to that of a weighted interpolation method based on zeros of Cheb...