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I read this question, which got me thinking. If a set of four types has no weaknesses, that means that for every two-type subset, its weaknesses are all resisted by the other two types. In case you're dying to know and too lazy to click on the link, the combinations are water ground flying steel and water bug steel dragon. This could be useful for building defensive cores, as they're supposed to consist of two Pokemon that are each able to switch into attacks that threaten the other. I chose sets of two Pokemon that are both viable in the same Smogon tier. There might actually be more such cores, because these weaknesses were calculated without considering abilities. Which of these might actually work?

Set in Uber (not technically a tier, but this post uses "tier" and "format" interchangeably)

Palkia @ Life Orb
Ability: Pressure
EVs: 4 Def / 252 SpA / 252 Spe
Timid Nature
-Spatial Rend
-Ice Beam

Genesect @ Choice Scarf
Ability: Download
EVs: 252 Atk / 4 Def / 252 Spe
Adamant Nature (hasty if adamant is impossible)
-Iron Head
-Extreme Speed
-Blaze Kick

Set in UU

Empoleon @ Life Orb
Ability: Torrent
EVs: 4 Def / 252 SpA / 252 Spe
Modest Nature
-Flash Cannon
-Grass Knot

Gligar @ Eviolite
Ability: Immunity
EVs: 252 HP / 4 Def / 252 SpD
Careful Nature

Set in NU

Steelix @ Leftovers
Ability: Sturdy
EVs: 252 HP / 4 Atk / 252 SpD
Sassy Nature
IVs: 0 Spe
-Gyro Ball
-Rock Slide

Mantine @ Leftovers
Ability: Water Absorb
EVs: 252 HP / 252 Def / 4 SpD
Bold Nature
-Air Slash

I'm not going to answer the question, but I need to teach you something about cores. Cores don't work based on covering each other's types - they work by covering each others threats. While this normally includes resisting each other's weaknesses, type charts aren't the end all and be all. Cores can function by having excellent offensive synergy (i.e beating the threats of your partner Pokemon via offensive pressure, momentum, luring and forcing switches) opposed to defensive synergy (covering each other's types to take as little damage as possible).

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This is actually an extremely popular and effective way of building defensive cores in Singles. You may have heard of "Dragon-Steel-Fairy" cores, which is arguably one of the most popular and synergistic defensive cores these days due to the fact that they check each other weaknesses, plus potent threats in the current meta, very well. While there are other things to consider, such as threats in the meta that's being played in, this is a great way to start.

Another thing to take into account when building defensive cores is the stats of the Pokemon. For example, the Empoleon/Gligar core is the best you mentioned for that reason. Since Gligar has high Defense, it can easily take Fighting type moves (which are usually Physical) aimed at Emploleon. Similarly, Empoleon's high Special Defense allows it to easily take Water and Ice type moves (which are usually Special) aimed at Gligar.

The one nitpick I have about the sets you listed is this: when building defensive cores, you want to choose Pokemon which have defensive sets. This is so they can continually absorb hits without going down. While the synergy might give offensive Pokemon some longevity, defensive Pokemon fill this role much better. I only say this because Palkia, Empoleon and Genesect seem to have offensive movesets.

One of my favorite cores consists of Mega-Scizor, Azumarill, and Garchomp:

  • 4x weak to Fire

Azumarill 2x resists Fire, Garchomp 2x resists Fire

  • 4x weak to Ice, 2x weak to Dragon, 2x weak to Fairy

Azumarill 2x resists Ice and immune to Dragon, Scizor 2x resists Ice, Dragon and Fairy

  • 2x weak to Poison, 2x weak to Electric, 2x weak to Grass

Scizor 4x resists Grass and immune to Poison, Garchomp 2x resists Poison and immune to Electric

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