Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userStability theorems for some Kruskal-Katona type results
92
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The classical Kruskal-Katona theorem gives a tight upper bound for the size of an $r$-uniform hypergraph $mathcal{H}$ as a function of the size of its shadow. Its stability version was obtained by Keevash who proved that if the size of $mathcal{H}$ is close to the maximum, then $mathcal{H}$ is structurally close to a complete $r$-uniform hypergraph. We prove similar stability results for two classes of hypergraphs whose extremal properties have been investigated by many researchers: the cancellative hypergraphs and hypergraphs without expansion of cliques.
rate research
Read More
The $chi$-stability index ${rm es}_{chi}(G)$ of a graph $G$ is the minimum number of its edges whose removal results in a graph with the chromatic number smaller than that of $G$. In this paper three open problems from [European J. Combin. 84 (2020) 103042] are considered. Examples are constructed which demonstrate that a known characterization of $k$-regular ($kle 5$) graphs $G$ with ${rm es}_{chi}(G) = 1$ does not extend to $kge 6$. Graphs $G$ with $chi(G)=3$ for which ${rm es}_{chi}(G)+{rm es}_{chi}(overline{G}) = 2$ holds are characterized. Necessary conditions on graphs $G$ which attain a known upper bound on ${rm es}_{chi}(G)$ in terms of the order and the chromatic number of $G$ are derived. The conditions are proved to be sufficient when $nequiv 2 pmod 3$ and $chi(G)=3$.
In this paper, we characterize the extremal digraphs with the maximal or minimal $alpha$-spectral radius among some digraph classes such as rose digraphs, generalized theta digraphs and tri-ring digraphs with given size $m$. These digraph classes are denoted by $mathcal{R}_{m}^k$, $widetilde{boldsymbol{Theta}}_k(m)$ and $INF(m)$ respectively. The main results about spectral extremal digraph by Guo and Liu in cite{MR2954483} and Li and Wang in cite{MR3777498} are generalized to $alpha$-spectral graph theory. As a by-product of our main results, an open problem in cite{MR3777498} is answered. Furthermore, we determine the digraphs with the first three minimal $alpha$-spectral radius among all strongly connected digraphs. Meanwhile, we determine the unique digraph with the fourth minimal $alpha$-spectral radius among all strongly connected digraphs for $0le alpha le frac{1}{2}$.
We study questions in incidence geometry where the precise position of points is `blurry (e.g. due to noise, inaccuracy or error). Thus lines are replaced by narrow tubes, and more generally affine subspaces are replaced by their small neighborhood. We show that the presence of a sufficiently large number of approximately collinear triples in a set of points in d dimensional complex space implies that the points are close to a low dimensional affine subspace. This can be viewed as a stable variant of the Sylvester-Gallai theorem and its extensions. Building on the recently found connection between Sylvester-Gallai type theorems and complex Locally Correctable Codes (LCCs), we define the new notion of stable LCCs, in which the (local) correction procedure can also handle small perturbations in the euclidean metric. We prove that such stable codes with constant query complexity do not exist. No impossibility results were known in any such local setting for more than 2 queries.
A {it superpattern} is a string of characters of length $n$ that contains as a subsequence, and in a sense that depends on the context, all the smaller strings of length $k$ in a certain class. We prove structural and probabilistic results on superpatterns for {em preferential arrangements}, including (i) a theorem that demonstrates that a string is a superpattern for all preferential arrangements if and only if it is a superpattern for all permutations; and (ii) a result that is reminiscent of a still unresolved conjecture of Alon on the smallest permutation on $[n]$ that contains all $k$-permutations with high probability.
Determination of the range of a variety of social choice correspondences: Plurality voting, the Borda rule, the Pareto rule, the Copeland correspondence, approval voting, and the top cycle correspondence
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
