A famous conjecture of Gyárfás and Sumner states for any tree T and integer k, if the chromatic number of a graph is large enough, either the graph contains a clique of size k or it contains T as an induced subgraph. We discuss some results and open problems about extensions of this conjecture to oriented graphs. We conjecture that for every oriented star S and integer k, if the chromatic numbe...

The Jacobian Conjecture can be generalized as follows: Let S be a polynomial ring of finitely many variables over a field of characterisitic zero and let T be a finitely generated extension domain of S .with T = k. If T is unramified over S, then T = S. The Jacobian Conjecture is the following : If f1, · · · , fn be elements in a polynomial ring k[X1, · · · , Xn] over a field k of characteristi...

We study pointwise convergence of the solutions to Schrödinger equations with initial datum f ∈ H(R). The conjecture is that the solution ef converges to f almost everywhere for all f ∈ H(R) if and only if s ≥ 1/4. The conjecture is known true for one spatial dimension and the convergence when s > 1/2 was verified for n ≥ 2. Recently, concrete progresses have been made in R for some s < 1/2. Ho...

We propose the following conjecture to generalize results of Pósa and Corrádi Hajnal. Let r, s be nonnegative integers and let G be a graph with |V (G)| ≥ 3r + 4s and minimal degree δ(G) ≥ 2r + 3s. Then G contains a collection of r + s vertex disjoint cycles, s of them with a chord. We prove the conjecture for r = 0, s = 2 and for s = 1. The corresponding extremal problem, to find the minimum n...

Erdős and Lovász conjectured in 1968 that for every graph G with χ(G) > ω(G) and any two integers s, t ≥ 2 with s + t = χ(G) + 1, there is a partition (S, T ) of the vertex set V (G) such that χ(G[S]) ≥ s and χ(G[T ]) ≥ t. Except for a few cases, this conjecture is still unsolved. In this note we prove the conjecture for quasi-line graphs and for graphs with independence number 2. AMS Subject C...

A linear ball is a simplicial complex whose geometric realization is homeomorphic to a ball and whose Stanley–Reisner ring has a linear resolution. It turns out that the Stanley–Reisner ring of the sphere which is the boundary complex of a linear ball satisfies the multiplicity conjecture. A class of shellable spheres arising naturally from commutative algebra whose Stanley–Reisner rings satisf...

graham higman has defined coset diagrams for psl(2,ℤ). these diagrams are composed of fragments, and the fragments are further composed of two or more circuits. q. mushtaq has proved in 1983 that existence of a certain fragment γ of a coset diagram in a coset diagram is a polynomial f in ℤ[z]. higman has conjectured that, the polynomials related to the fragments are monic and for a fixed degree...

چکیده the issue of validity and non-validity of absolute conjecture, although frequently studied and researched by great scholars, is still in need of further research, since the studies and researches carried out so far are typically based on the presuppositions that have been regarded as indisputable and needless of study and research; whereas, in fact, they are not indisputable and need furt...

Given a function f : {0, 1} n → R, its Fourier Entropy is de ned to be −∑S f̂(S) log f̂(S), where f̂ denotes the Fourier transform of f. This quantity arises in a number of applications, especially in the study of Boolean functions. An outstanding open question is a conjecture of Friedgut and Kalai (1996), called Fourier Entropy In uence (FEI) Conjecture, asserting that the Fourier Entropy of any ...

Let A1,…,Am be families of k-subsets an n-set. Suppose that one cannot choose pairwise disjoint edges from s+1 distinct families. Subject to this condition we investigate the maximum |A1|+…+|Am|. Note subcase m=s+1, A1=…=Am is Erdős Matching Conjecture, most important open problems in extremal set theory. We provide some upper bounds, a general conjecture and its solution for range n≥4k2s.