Linux has a (configurable) limit on the neighbor table. I know in RHEL variants, the default has been 1024 neighbors for a while.

net.ipv4.neigh.default.gc_thresh3
net.ipv4.neigh.default.gc_thresh2
net.ipv4.neigh.default.gc_thresh1

net.ipv6.neigh.default.gc_thresh3
net.ipv6.neigh.default.gc_thresh2
net.ipv6.neigh.default.gc_thresh1

These may be rough guidelines for performance or arbitrary limits someone thought would be a good idea. Either way, you'll need to increase the number if you're using IP on Linux.

Although not explicitly stated, I would assume that these computers may be virtualized or inside some sort of blade chassis (which reduces the number of physical cables to a switch). Strictly speaking, I see no hardware limitation in your way, as most top of rack switches will easily do a few thousand or 10's of thousands of MAC entries and a few thousand hosts can fit inside a single IP4 or IP6 subnet. There are some pretty dense switches if you actually do need 1000 ports, but as others have stated, you'll utilize a good portion of the rack in cable and connectors.

--Blake

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