if field free is a bool (tinyInt), you could try something like this:
SELECT * FROM (
SELECT `hotelName`, MIN(`available`) AS _available FROM `hotels` WHERE dbdate BETWEEN 20130507 AND 20130511 GROUP BY `hotelName`)a
WHERE _available;
The above query gave both hotels from the test as a result, when you change 20130507 to 20130503, it will only return the Amsterdam Hotel.
with this as a test table:
CREATE TABLE `hotels` (
`hotelName` varchar(20) DEFAULT NULL,
`dbdate` int(11) DEFAULT NULL,
`available` tinyint(1) DEFAULT NULL
) ENGINE=MyISAM DEFAULT CHARSET=latin1;
/*Data for the table `hotels` */
insert into `hotels`(`hotelName`,`dbdate`,`available`) values ('Amsterdam Hotel',20130503,1),('Amsterdam Hotel',20130504,1),('Amsterdam Hotel',20130505,1),('Amsterdam Hotel',20130506,1),('Amsterdam Hotel',20130507,1),('Amsterdam Hotel',20130508,1),('Amsterdam Hotel',20130509,1),('Amsterdam Hotel',201305010,1),('Amsterdam Hotel',201305011,1),('Amsterdam Hotel',201305012,1),('Utrecht Hotel',20130503,0),('Utrecht Hotel',20130504,0),('Utrecht Hotel',20130505,0),('Utrecht Hotel',20130506,0),('Utrecht Hotel',20130507,1),('Utrecht Hotel',20130508,1),('Utrecht Hotel',20130509,1),('Utrecht Hotel',201305010,1),('Utrecht Hotel',201305011,1),('Utrecht Hotel',201305012,0);