What should the spacing be between plugs when core aerating a lawn?

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Here is some basic information and fact sheets from several reliable sources about tine placement and when to aerate depending on what type of grass you have.

Tines are usually spoon-shaped or hollow tubes. As the tubes are driven into the lawn, cores of soil are removed from the ground and strewn across the lawn. Both types of tines work equally well, but the hollow tine makes a somewhat cleaner hole than the spoon type and brings up less soil. The tine size varies from one-fourth of an inch up to three-fourth of an inch and can penetrate the soil to a depth of 2 to 3 inches, depending on the manufacturer’s specifications. Tines should be spaced between 2 and 3 inches. A closer tine placement on the aerator removes more soil, exposes more soil surface area for water and fertilizer movement, and alleviates compaction better than a wider tine spacing.

The penetration depth depends on soil type, soil moisture, tine diameter, and the weight and power of the aerifier. Soil cores should be left on the lawn to be broken up by rainfall and traffic. If their appearance bothers you, you can speed up their disappearance by raking them into the grass. Whichever machine you use, go over the lawn twice, once in one direction, and then in a perpendicular direction for best results.

Aerification can be combined with seeding, particularly on sparse or bare areas. If you are going to seed the lawn, you should make six to 10 passes over the area with a machine. You need to produce a number of holes, at least 4 inches apart, to improve the appearance and density of the stand. Allow the holes about a month to heal before seeding. If you overseed immediately after coring, seeds that land in or near the aerifier holes will germinate and grow much better than those between the holes, giving the lawn an uneven, speckled appearance. With a fraction of the effort and expense of tilling up the entire area, combining aerification with seeding will give the lawn a brand-new look.

https://hgic.clemson.edu/factsheet/aerating-lawns/

https://extension.psu.edu/aeration-of-turfgrass-areas

https://hyg.ipm.illinois.edu/article.php?id=856