Seminarium Kombinatoryka, Teoria Grafów i Zbiorów Uporządkowanych
Wydział Matematyki i Nauk Informacyjnych, Politechnika Warszawska
Kalendarz Środy, 10:15-11:45
Pinezka Gmach MiNI PW, Sala 431, ul. Koszykowa 75, Warszawa
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Najbliższy referat
11.06.2025 Michał Dybowski PW
Chessboard and level sets of continuous functions
\(\quad\)Let some segments of the chessboard be mined. Assume that the king cannot go across the chessboard from the left edge to the right one without meeting a mined square. Then the rook can go from upper edge to the lower one moving exclusively on mined segments. This theorem was generalized by Tkacz and Turzański to the n-dimensional chessboard in two versions. One of them states: Let each segment of the n-dimensional chessboard be colored with one of the n colors. Then there exist pair of opposite faces of the chessboard and connected chain of monochromatic segments which connects these faces. They also showed that the Poincaré–Miranda Theorem and its parametric extension are consequences of their result. In this talk we shall consider a similar problem to the n-dimensional Steinhaus Chessboard Theorem with some restrictions on the arrangement of colors. The obtained result is related with that of Tkacz and Turzański, and the proof employs their result along with the concept of clustered chromatic number which comes from the graph theory. Furthermore, we will discuss certain optimal constants associated with this problem. Additionally, we introduce a generalization of the clustered chromatic number, wherein the fixed distance between monochromatic connected components is treated as a variable parameter. The consequence of the result is some Poincaré–Miranda type theorem. It turns out that this new result can be applied to prove the n-dimensional Steinhaus Chessboard Theorem with relative ease, implying that these results are equivalent. Independently, we can apply the result to prove the Brouwer Fixed Point Theorem which is equivalent to the Poincaré–Miranda Theorem.
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