@article{22158,
  abstract     = {The triangle removal states that if G contains  edge-disjoint triangles, then G contains  triangles. Unfortunately, there are no sensible bounds on the order of growth of , and at any rate, it is known that  is not polynomial in . Csaba recently obtained an asymmetric variant of the triangle removal, stating that if G contains  edge-disjoint triangles, then G contains  copies of . To this end, he devised a new variant of Szemerédi’s regularity lemma. We obtain the following results:

• We first give a regularity-free proof of Csaba’s theorem, which improves the number of copies of  to the optimal number .

• We say that H is -abundant if every graph containing  edge-disjoint triangles has  copies of H. It is easy to see that a -abundant graph must be triangle-free and tripartite. Given our first result, it is natural to ask if all triangle-free tripartite graphs are -abundant. Our second result is that assuming a well-known conjecture of Ruzsa in additive number theory, the answer to this question is negative.

Our proofs use a mix of combinatorial, number-theoretic, probabilistic and Ramsey-type arguments.},
  author       = {Gishboliner, Lior and Shapira, Asaf and Wigderson, Yuval},
  issn         = {2050-5094},
  journal      = {Forum of Mathematics, Sigma},
  publisher    = {Cambridge University Press},
  title        = {{An efficient asymmetric removal lemma and its limitations}},
  doi          = {10.1017/fms.2024.68},
  volume       = {13},
  year         = {2025},
}

@article{22161,
  abstract     = {Recently, Souza introduced blowup Ramsey numbers as a gener-
alization of bipartite Ramsey numbers. For graphs G and H, say
G r
−→ H if every r-edge-coloring of G contains a monochromatic
copy of H. Let H[t] denote the t-blowup of H. Then the blowup
Ramsey number of G, H, r, and t is defined as the minimum n
such that G[n] r
−→ H[t]. Souza proved upper and lower bounds on
n that are exponential in t, and conjectured that the exponential
constant does not depend on G. We prove that the dependence on
G in the exponential constant is indeed unnecessary, but conjecture
that some dependence on G is unavoidable.
An important step in both Souza’s proof and ours is a theorem of
Nikiforov, which says that if a graph contains a constant fraction
of the possible copies of H, then it contains a blowup of H of
logarithmic size. We also provide a new proof of this theorem with
a better quantitative dependence.},
  author       = {Fox, Jacob and Luo, Sammy and Wigderson, Yuval},
  issn         = {2150-959X},
  journal      = {Journal of Combinatorics},
  number       = {1},
  pages        = {1--15},
  publisher    = {International Press of Boston},
  title        = {{Extremal and Ramsey results on graph blowups}},
  doi          = {10.4310/joc.2021.v12.n1.a1},
  volume       = {12},
  year         = {2021},
}