Powers of Hamilton Cycles of High Discrepancy are Unavoidable

The Pósa-Seymour conjecture asserts that every graph on n vertices with minimum degree at least (1−1/(r +1))n contains the r-th power of a Hamilton cycle. Komlós, Sárközy and Szemerédi famously proved for large n. notion discrepancy appears in many areas mathematics, including theory. In this setting, G is given along 2-coloring its edges. One then asked to find copy subgraph discrepancy, i.e., significantly more than half edges one color. For r > 2, we determine threshold needed cycle answering question posed by Balogh, Csaba, Pluhár Treglown. Notably, 3, approximately matches requirement conjecture.