A tower lower bound for the degree relaxation of the Regularity Lemma

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985807%3A_____%2F25%3A00645680" target="_blank" >RIV/67985807:_____/25:00645680 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.5070/C65465674" target="_blank" >https://doi.org/10.5070/C65465674</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.5070/C65465674" target="_blank" >10.5070/C65465674</a>

Result language
angličtina
Original language name
A tower lower bound for the degree relaxation of the Regularity Lemma
Original language description
It is well-known that if (A, B) is an ε/2 -regular pair (in the sense of Szemeredi) then there exist sets A′ ⊆ A and B′ ⊆ B with |A′ | ⩽ ε|A| and |B′ | ⩽ ε|B| so that the degrees of all vertices in AA′ differ by at most ε|B| and the degrees of all vertices in BB′ differ by at most ε|A|. We call such a property ε-degularity. This leads to the notion of an ε-degular partition of a graph in the same way as the definition of ε-regular pairs leads to the notion of ε-regular partitions. We show that there exist graphs in which any ε-degular partition requires the number of clusters to be tower(Θ(ε^(−1/3))). That is, even though degularity is a substantial relaxation of regularity, in general one cannot improve much on the bounds that come with Szemeredi’s regularity lemma.
Czech name
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Czech description
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Project
<a href="/en/project/GX21-21762X" target="_blank" >GX21-21762X: Graph limits and beyond</a><br>
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2025
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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