positive integer $n>k>t$ let $\binom{[n]}{k}$ denotes a collection of all $k$ subsets of the standard $n$ element set $[n]=\{1,\ldots,n\}$. a subset of $\binom{[n]}{k}$ is called $k$-graph. If $|F\cap F’|\geq t$ for all $F,F’\in \mathcal{F}$ then $k$-graph $\mathcal{F}$ is $t$- called intersecting. One of the central results of extreme set theory is the Erd\H{o}s-Ko-Rado theorem. $k$-graph has more than $\binom{nt}{kt}$ edges. If $n$ is greater than this threshold, $t$-star (all $k$-sets containing a fixed $t$-set) is the only family that achieves this bound. Define $\mathcal{F}(i)=\{F\setminus \{i\}\colon i\in F\in \mathcal{F}\}$ . The quantity $\varrho(\mathcal{F})=\max\limits_{1\leq i\leq n}|\mathcal{F}(i)|/|\mathcal{F}|$ is $ k$- The graph is a star. The main result (Theorem 1.5) is that $\varrho(\mathcal{F})>1/d$ is shown to hold. ^{2d+1}\binom{nd-1}{kd-1}$ and $n\geq 4(d-1)dk$. Such statements are \cite{F78-2} and \cite{ DF} results, but only for much larger values ​​of $n/k$ and/or $n$. The proof is purely combinatorial and based on a new method: shift ad extremis. Applying the same method, we obtain some near-optimal bounds for the case of $t\geq 2$ (Theorem 1.11) and many related results.

    Source link


    Leave A Reply