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

EMILIANO J.J. ESTRUGO; ADRIÁN PASTINE

Arco

Conferencia; Combinatorics; 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 \cite{Kap}, Konvalina \cite{Kon}, and Mansour and Sun \cite{Man}, 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 \cite{Tal} 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.\bibitem{Kap}I. Kaplansky, Solution of the probl\`{e}me des m\`{e}nages, {\em Bull. Amer. Math. Soc.} {\bf 49} (1943), 784-785.\bibitem{Kon}J. Konvalina, On the number of combinations without unit separation, {\em J. Combin. Theory Ser. A} {\bf 31} (1981), 101-107.\bibitem{Man}T. Mansour and Y. Sun, On the number of combinations without certain separations, {\em European J. Combin.} (2007), 1200-1206.\bibitem{Tal}J. Talbot, Intersecting families of separated sets, {\em J. Lond. Math. Soc.} {\bf 68} (1) (2003), 37-51.