What is the minimum number of chess pieces of a given type needed to attack every vacant square of the board? The answer turns out to be 9 for kings, 5 for queens, 8 for bishops or rooks, and 12 for knights. The picture above shows some minimal solutions to the problem for kings, queens, bishops, and knights, and the placement shown for the bishops also works for rooks. Problems of this sort are called
Hi Varun! The answer is “yes”. My understanding is that Roth’s Theorem proves that the size of a Salem—Spencer set is almost, but not quite linear in n (see the Wikipedia page on Salem—Spencer sets). Roth’s Theorem and Van der Waerden’s Theorem are both special cases of Szemerédi’s Theorem.
Szemerédi visited Boulder once as the DeLong lecturer, before you were here. He was an entertaining speaker.
Do the Salem–Spencer sets have any application or relation to Van der Waerden's theorem and Szemeredi's theorem?
Hi Varun! The answer is “yes”. My understanding is that Roth’s Theorem proves that the size of a Salem—Spencer set is almost, but not quite linear in n (see the Wikipedia page on Salem—Spencer sets). Roth’s Theorem and Van der Waerden’s Theorem are both special cases of Szemerédi’s Theorem.
Szemerédi visited Boulder once as the DeLong lecturer, before you were here. He was an entertaining speaker.
I really enjoyed the connection with chess! Thank you for another post to learn from 👍👍😄