Multicolor $K_r$-Tilings with High Discrepancy
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866911556553408512 |
|---|---|
| author | Chan, Henry Cheng, Daniel Gishboliner, Lior Li, Xiangyu |
| author_facet | Chan, Henry Cheng, Daniel Gishboliner, Lior Li, Xiangyu |
| contents | We study the minimum degree threshold $δ_{r,q}$ guaranteeing the existence of $K_r$-tilings of high discrepancy in any $q$-edge-coloring. Balogh, Csaba, Pluhár and Treglown handled the 2-color case, proving that $δ_{r,2} = \frac{r}{r+1}$ for all $r \geq 3$. Here we determine $δ_{r,q}$ for all $q$ large enough, namely $q \geq \binom{r}{2}$. For example, we show that for $r \geq 4$, $δ_{r,q} = \frac{r}{r+1}$ for $\binom{r}{2} \leq q \leq \binom{r+1}{2}$ and $δ_{r,q} = \frac{r-1}{r}$ for $q \geq \binom{r+1}{2}+2$. Thus, $δ_{r,q}$ has a phase transition at $q = \binom{r+1}{2}$, where it drops from $\frac{r}{r+1}$ and then stabilizes at the existence threshold $\frac{r-1}{r}$. We also show that $δ_{r,q} \leq \frac{r}{r+1}$ for all $r,q$, supplementing and giving a new proof for the result of Balogh, Csaba, Pluhár and Treglown. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_29277 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Multicolor $K_r$-Tilings with High Discrepancy Chan, Henry Cheng, Daniel Gishboliner, Lior Li, Xiangyu Combinatorics We study the minimum degree threshold $δ_{r,q}$ guaranteeing the existence of $K_r$-tilings of high discrepancy in any $q$-edge-coloring. Balogh, Csaba, Pluhár and Treglown handled the 2-color case, proving that $δ_{r,2} = \frac{r}{r+1}$ for all $r \geq 3$. Here we determine $δ_{r,q}$ for all $q$ large enough, namely $q \geq \binom{r}{2}$. For example, we show that for $r \geq 4$, $δ_{r,q} = \frac{r}{r+1}$ for $\binom{r}{2} \leq q \leq \binom{r+1}{2}$ and $δ_{r,q} = \frac{r-1}{r}$ for $q \geq \binom{r+1}{2}+2$. Thus, $δ_{r,q}$ has a phase transition at $q = \binom{r+1}{2}$, where it drops from $\frac{r}{r+1}$ and then stabilizes at the existence threshold $\frac{r-1}{r}$. We also show that $δ_{r,q} \leq \frac{r}{r+1}$ for all $r,q$, supplementing and giving a new proof for the result of Balogh, Csaba, Pluhár and Treglown. |
| title | Multicolor $K_r$-Tilings with High Discrepancy |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.29277 |