Gallai-type Results for Multiple Boxes and Forests

J. Lehel
1988 European journal of combinatorics (Print)  
Gallai-type statements considered here have the next general form: ifF is a finite family of subsets of some underlying space such that every k members (k ;;. 2) have a common point then there exist t points meeting every member of F. Concerning the interplay of the parameters k and t, we obtain results for two generalizations of multiple interval structures: ford-dimensional multiple boxes and for subforests of an underlying tree. In case of families composed by the unions of c d-dimensional
more » ... xes, we prove that the minimum value oft is finite for fixed c, dand k, iff k ;;. min {c, d} + l. Incase of families composed by c-component subforests of a tree, we prove that the minimum value oft is equal to c and t = c is attainable for k = c + l. Foreveryintegerc,dandk;;;. min{c, d} + 1, thereexistsaconstantt(dependingonly on k, c and d) with the property: if Fis any family of d-dimensional c-boxes such that every k members have a common point then there are t points meeting every member of F. This statement is proved in section 2 by slightly generalizing the proof of an analogous result for c-component forests of a tree (in [6]). Just as in the case of forests, we obtain a more general Gallai-type theorem (Theorem 2.2) immediately implying our result for multiple boxes (Theorem 2.3). In section 3 we make simple observations on subtree families of an underlying tree, all of which follow immediately by the linear ordering of the points when we restrict ourselves 113 0195-6698/88/020113 + 08 $02.00/0
doi:10.1016/s0195-6698(88)80035-0 fatcat:slc3u6lxk5hhliagsbbtsoskem