Say you have n files called 1.bed through n.bed.

You can pre-process your n files to make them four-column BED files, where the fourth column identifies their source filename:

$ for idx in `seq 1 n`; do cut -f1-3 $idx.bed | awk -vidx=$idx '{ print $0"\tfile-"idx; }' - > $idx.id.bed; done

This approach with BEDOPS bedmap --count generalizes to n internally-disjoint input BED4 files, in which the fourth column contains a string identifying the filename:

$ bedops --everything 1.id.bed 2.id.bed 3.id.bed ... n.id.bed \
    | bedmap --echo --count --echo-map-id --delim '\t' - \
    > answer.bed

The bedmap step uses, by default, overlap of one or more bases for inclusion. You can modify this threshold to be more stringent.

The file answer.bed has six columns. The first three columns are the genomic interval. The fourth column is the ID value indicating which file the interval comes from. The fifth column indicates the number of overlaps with all the intervals from all n files. The sixth column is a semi-colon delimited string indicating the files where overlaps are occurring.