This answer was based on the ideas from here and here. 
The idea of the algorithm is the following:
- We create an array where the indexes are all the possible minutes for a date and the values are - true.
 
- We fill all the minutes which are included on the existing events to - false.
 
- Finally, we look for consecutive time intervals of size - $length(in minutes) and once we can traverse the array without finding a- falsevalue for- $lengthiterations, we found an empty timeslot for a new event.
 
<?php
require("database.php");
// Create an array with all the timeslots (in minutes) of a date
// Example: $date = '2018-03-21'
//          $timeslots['2018-03-21 00:00:00'] = true;
//          $timeslots['2018-03-21 00:01:00'] = true;
//          $timeslots['2018-03-21 00:02:00'] = true;
//                            until
//          $timeslots['2018-03-21 23:59:00'] = true;
function getAllTimeslots($date)
{
    $currentDate = strtotime($date . " 00:00:00");
    $endDate = strtotime($date . " 23:59:59");
    $timeslots = [];
    while ($currentDate <= $endDate) {
        $index = date("Y-m-d H:i:s", $currentDate);
        $timeslots[$index] = true;
        $currentDate = strtotime("+1 minute", $currentDate);
    }
    return $timeslots;
}
// Based on the events table registers, fill the intervals of the timeslots array marking them as false, i.e., not available
// Therefore, it will result on an array with trues and falses marking if the specific minute is already filled by an event
function fillUsedTimeslots($date, $timeslots)
{
    $queryDate = $date . "%";
    $conn = openConnection();
    $result = $conn->query("SELECT dt_start, dt_end
                            FROM events
                            WHERE dt_start LIKE '" . mysqli_real_escape_string($conn, $queryDate) . "'
                                OR dt_end LIKE '" . mysqli_real_escape_string($conn, $queryDate) . "'");
    if ($result->num_rows > 0) {
        while ($row = $result->fetch_assoc()) {
            $start = strtotime($row['dt_start']);
            $end = strtotime($row['dt_end']);
            // update the timeslots data structure with the information of the events' time usage
            while ($start < $end) {
                $currentDatetime = date("Y-m-d H:i:s", $start);
                $timeslots[$currentDatetime] = false;
                $start = strtotime("+1 minute", $start);
            }
        }
    }
    return $timeslots;
}
// Finally, we need to find a free an interval of time ($length) where we can place an event
// In short words, it iterates over the array of $timeslots looking for $length consecutives trues, i.e.,
// if $length=60, we iterate over the array looking for 60 times of consecutives falses
function findFreeTimeslots($date, $length, $timeslots)
{
    $currentDate = strtotime($date . " 00:00:00");
    $endDate = strtotime($date . " 23:59:00");
    $timeInterval = 0;
    while ($currentDate <= $endDate) {
        $index = date("Y-m-d H:i:s", $currentDate);
        if ($timeslots[$index]) { // Timeslot is free for use
            $timeInterval += 1;
        } else { // Reset timeInterval
            $timeInterval = 0;
        }
        // We have $length consecutives true, i.e., an interval of $length minutes available for another event
        if ($timeInterval == $length + 1) {
            echo "<br/>Timeslot found: " . date("Y-m-d H:i:s", strtotime("-" . $length . " minutes", $currentDate)) . " - " . $index;
            $timeInterval = 1;
        }
        $currentDate = strtotime("+1 minute", $currentDate);
    }
}
// Main
$timeslots = getAllTimeslots("2013-09-21");
$filledTimeslots = fillUsedTimeslots("2013-09-21", $timeslots);
findFreeTimeslots("2013-09-21", 180, $filledTimeslots);
I applied your example of 3 hours interval, and the result was:
Timeslot found: 2013-09-21 00:00:00 - 2013-09-21 03:00:00
Timeslot found: 2013-09-21 03:00:00 - 2013-09-21 06:00:00
Timeslot found: 2013-09-21 16:30:00 - 2013-09-21 19:30:00
Timeslot found: 2013-09-21 19:30:00 - 2013-09-21 22:30:00
You can restrict the timeslots data structure size to only working times, or a diferent criteria to adapt on your project. And also, refactor it, it's been a long time since  I programmed vanilla PHP