Consider each person in the party to be a point in the plane of $1000$ points. Construct concentrically placed hexagons by adding points around the perimeter of the hexagonal grid until you use most of the $1000$ points. (Refer to figure attached.) Each point in the interior (excluding the points on the perimeter) of the hexagonal grid is equidistant to exactly $6$ other points. Consider the lines joining each point to the 6 equidistant neighboring points to be acquitanceships. Clearly there are $1 + 6*1 + 6*2 + 6*3 + ... 6*17 = 919$ points in this grid. There are $81$ points remaining in the plane. Out of the $919$ points above all points excluding the perimeter $= 919 - 6*17 = 817$ have exaclty $6$ acqauintances. Out of the $81$ points, $17*4$ points are needed to construct hexagons on $4$ sides of the main grid so that $16$ points on each of these 4 sides of the grid are now centers of new hexagons. On the $5$th side of the grid, we could use the remining $13$ points to form $12$ new hexagons. So total number of points on the perimeter of the grid that are now surrounded by $6$ equidistant points is $= 16*4 + 12 = 76$. So total number of points with exactly $6$ acquaintacnes is $817 + 76 = 893$. Thus is is possible to find atleast $890$ people who are acquainted with exactly $6$ other guests.