An Erdös-Ko-Rado type theorem for 1-separated k-sets in several circles

EMILIANO J.J. ESTRUGO; ADRIÁN PASTINE

Río de Janeiro

Congreso; International Congress of Mathematicians; 2018

A set of $k$ ordered objects that does not contain elements at certain distances $S=\{d_1,d_2,\ldots,d_s\}$ is called \textit{$S$-separated $k$-set}. Kaplansky (1943), Konvalina (1981), and Mansour and Sun (2007), studied how many separated $k$-sets are ordered in different ways. Kaplansky studied $\{1\}$-separeted $k$-sets ordered in a circle or line. Konvaline studied the amount of $\{1\}$-separated $k$-sets in two circles or lines of the same size. Mansour and Sun studied how many $\{m-1,2m-1,\ldots,pm-1\}$-separated sets are in a circle or line. In (2003) Talbot proved an Erd\"{o}s-Ko-Rado type theorem for intersecting families of $\{1,\ldots,s\}$-separated $k$-sets in a circle. This is a Theorem that studies the size of maximum intersecting families, and characterize them. In this work, we study an Erd\"{o}s-Ko-Rado type theorem for $\{1\}$-separated intersecting families and the amount of $\{1\}$-separated $k$-sets in several circles of different sizes.