Generalized Ramsey Numbers in the Hypercube
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914272123027456 |
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| author | Heath, Emily Schwieder, Coy Zerbib, Shira |
| author_facet | Heath, Emily Schwieder, Coy Zerbib, Shira |
| contents | We study the generalized Ramsey numbers $f(Q_n, C_{k}, q)$, that is, the minimum number of colors needed to edge-color the hypercube $Q_n$ so that every copy of the cycle $C_{k}$ has at least $q$ colors. Our main result is that for any integers $k,q$ satisfying $k \geq 6$ and $3 \leq q \leq k/2+1$, we have $f(Q_n, C_{k}, q)= o\left( n^{\frac{k/2-1}{k-q+1}} \right).$ We also prove a few other upper and lower bounds in the special cases $k=4$ and $k=6$. This continues the line of research initiated by Faudree, Gyárfás, Lesniak, and Schelp and Mubayi and Stading who studied the case $k=q$, and by Conder who considered the case $k=6$ and $q=2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_15451 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Generalized Ramsey Numbers in the Hypercube Heath, Emily Schwieder, Coy Zerbib, Shira Combinatorics We study the generalized Ramsey numbers $f(Q_n, C_{k}, q)$, that is, the minimum number of colors needed to edge-color the hypercube $Q_n$ so that every copy of the cycle $C_{k}$ has at least $q$ colors. Our main result is that for any integers $k,q$ satisfying $k \geq 6$ and $3 \leq q \leq k/2+1$, we have $f(Q_n, C_{k}, q)= o\left( n^{\frac{k/2-1}{k-q+1}} \right).$ We also prove a few other upper and lower bounds in the special cases $k=4$ and $k=6$. This continues the line of research initiated by Faudree, Gyárfás, Lesniak, and Schelp and Mubayi and Stading who studied the case $k=q$, and by Conder who considered the case $k=6$ and $q=2$. |
| title | Generalized Ramsey Numbers in the Hypercube |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2601.15451 |