Consulta de intervalos de fechas SQL

Si respondo bien la pregunta, con [:start, :end] siendo su rango de fechas de interés, está buscando:

select *
 from event
where -- event started earlier, ends later
      start <= :start and :start <= end
   or -- event starts during [:start, :end]
      :start <= start and start <= :end
   or -- event ends during [:start, :end]  
      :start <= end and end <= :end;

Si está buscando un :day en particular , usa :day como :start y :day + 1 day como :end .