Turán theorems for unavoidable patterns

We prove Turán-type theorems for two related Ramsey problems raised by Bollobás and by Fox and Sudakov. First, for t ≥ 3, we show that any two-colouring of the complete graph on n vertices that is δ-far from being monochromatic contains an unavoidable t-colouring when δ ≫ n−1/t, where an unavoidable t-colouring is any two-colouring of a clique of order 2t in which one colour forms either a clique of order t or two disjoint cliques of order t. Next, for t ≥ 3, we show that any tournament on n vertices that is δ-far from being transitive contains an unavoidable t-tournament when δ ≫ n−1/[t/2], where an unavoidable t-tournament is the blow-up of a cyclic triangle obtained by replacing each vertex of the triangle by a transitive tournament of order t. Conditional on a well-known conjecture about bipartite Turán numbers, both our results are sharp up to implied constants and hence determine the order of magnitude of the corresponding off-diagonal Ramsey numbers.

How Will Information Services of Social Media Portals Change in the Future When Artificial Intelligence is implemented?

Innovation is surpassing our desires. Different AI-based instruments are created and help advertisers focus on perfect clients via web-based media at the ideal time. Purpose: the purpose of the study is to link the Information Services of Social Media Portals Change in the Future When Artificial Intelligence is implemented. Methodology: Surveys and qualitative methodologies were utilized to build up the theoretical perspective on A.I.'s effects on advancement across various foundations and social areas on the planet's economies. All the users of face book were population for the current study. Convenient sampling was the technique to access the sample. Focus groups and academic centres, and governmental institutions were likewise fused in acquiring valuable data on the effects of being knowledgeable about all the fields of advancement through A.I. innovations on social media. Findings: Most interviewees reacted emphatically about using technologically advanced tools and disseminating information in modern applications on social media. Advancement in A.I. innovations in the social media services expanded creation periods in the viral news than anticipated from a man if they needed to access vital information at every interval without an aspect of delay. Contribution: The future will be A.I. empowered and less miss-information. Tis paper do not generally know how soon or in what ways these dreams of the AI-empowered future will turn into a reality, yet even the most easygoing of eyewitnesses can see that these are unavoidable patterns. The one sure thing is that it is coming, and this paper now travelling toward that path.

Note on Gy. Elekes's Conjectures Concerning Unavoidable Patterns in Proper Colorings

A counterexample is presented to Gy. Elekes's conjecture concerning the existence of long $2$-colored paths in properly colored graphs. A modified version of the conjecture is given and its connections to a problem of Erdős - Gyárfás and to Szemerédi's theorem are examined.