Rainbow Turán number of even cycles, repeated patterns and blow-ups of cycles

Given positive integers p and q, a p q ( , )-coloring of the complete graph K n is an edge-coloring in which every pclique receives at least q colors. Erdős and Shelah posed the question of determining f n p q ( , , ), the minimum number of colors needed for a p q ( , )-coloring of K n . In this paper, we expand on the color energy technique introduced by Pohoata and Sheffer to prove new lower bounds on this function, making explicit the connection between bounds on extremal numbers and f n p q ( , , ).Using results on the extremal numbers of subdivided complete graphs, theta graphs, and subdivided complete bipartite graphs, we generalize results of Fish, Pohoata, and Sheffer, giving the first nontrivial lower bounds on f n p q ( , , ) for some pairs p q ( , ) and improving previous lower bounds for other pairs.

show abstract

“…Applying (4) gives a result which matches Theorem 2.5 due to our current knowledge of the extremal number of

{K}_{t}^{+}

. Among other impressive results, Janzer [15] showed that for fixed integers

k,r\ge 2

{f}_{r}(n,{C}_{2k})={\rm{\Omega }}\left({n}^{\frac{r}{r-1}\cdot \frac{k-1}{k}}\right).

…”

Section: Introductionmentioning

confidence: 52%

Lower bounds on the Erdős–Gyárfás problem via color energy graphs

Balogh

English

Heath

et al. 2023

Journal of Graph Theory

View full text Add to dashboard Cite

show abstract

“…Whereas the recent improvement in [9] took a different approach to that of Janzer [5], our proof is in spirit closer to that of [5] using homomorphism counts and improves it in two ways. First, we use more efficient lopsided regularization lemma than the Jiang-Seiver lemma [7] used by Janzer.…”

Section: Introductionmentioning

confidence: 84%

“…It is shown that ex * (n, C) = Ω(n log n) in [8] and Das, Lee, and Sudakov [2] obtained an upper bound O(ne (log n) 1 2 +o(1) ). There have been some recent improvements upon the upper bound [5,9] and the current best one is O(n(log n) 2+o (1) ) appeared in [9]. We improve this bound to O(n log 2 n).…”

Section: Introductionmentioning

confidence: 92%

“…In [8], a general lower bound ex * (n, C 2k ) = Ω(n 1+1/k ) is obtained, whereas the matching upper bounds were only verified for k = 2, 3. This upper bound was subsequently improved by Das, Lee, and Sudakov [2] to O(n 1+(1+o k (1)) log k/k ) and by Janzer [5] to O(n 1+1/k ). While Janzer's bound matches the lower bound given in [8], the implicit constant is exponential in k. We improve it to a polynomial one as follows.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Rainbow cycles in properly edge-colored graphs

Kim¹,

Lee²,

Liu³

et al. 2022

Preprint

View full text Add to dashboard Cite

We prove that every properly edge-colored n-vertex graph with average degree at least 100(log n) 2 contains a rainbow cycle, improving upon (log n) 2+o(1) bound due to Tomon. We also prove that every properly colored n-vertex graph with at least 10 5 k 2 n 1+1/k edges contains a rainbow 2k-cycle, which improves the previous bound 2 ck 2 n 1+1/k obtained by Janzer.Our method using homomorphism inequalities and a lopsided regularization lemma also provides a simple way to prove the Erdős-Simonovits supersaturation theorem for even cycles, which may be of independent interest.

show abstract

“…The first nontrivial upper bound was obtained by Das, Lee and Sudakov [9], who showed that for any 𝛾 > 0 and sufficiently large n, we have 𝑓 (𝑛) ≤ 𝑛 exp((log 𝑛) 1/2+𝛾 ). Janzer [18] proved that 𝑓 (𝑛) = 𝑂 (𝑛(log 𝑛) 4 ). The current best bound is due to Tomon [28] who showed that 𝑓 (𝑛) ≤ 𝑛(log 𝑛) 2+𝑜 (1) .…”

Section: Rainbow Cyclesmentioning

confidence: 99%

On the Turán number of the hypercube

Janzer,

Sudakov

2024

Forum of Mathematics, Sigma

View full text Add to dashboard Cite

In 1964, Erdős proposed the problem of estimating the Turán number of the d-dimensional hypercube $Q_d$ . Since $Q_d$ is a bipartite graph with maximum degree d, it follows from results of Füredi and Alon, Krivelevich, Sudakov that $\mathrm {ex}(n,Q_d)=O_d(n^{2-1/d})$ . A recent general result of Sudakov and Tomon implies the slightly stronger bound $\mathrm {ex}(n,Q_d)=o(n^{2-1/d})$ . We obtain the first power-improvement for this old problem by showing that $\mathrm {ex}(n,Q_d)=O_d\left (n^{2-\frac {1}{d-1}+\frac {1}{(d-1)2^{d-1}}}\right )$ . This answers a question of Liu. Moreover, our techniques give a power improvement for a larger class of graphs than cubes. We use a similar method to prove that any n-vertex, properly edge-coloured graph without a rainbow cycle has at most $O(n(\log n)^2)$ edges, improving the previous best bound of $n(\log n)^{2+o(1)}$ by Tomon. Furthermore, we show that any properly edge-coloured n-vertex graph with $\omega (n\log n)$ edges contains a cycle which is almost rainbow: that is, almost all edges in it have a unique colour. This latter result is tight.

show abstract

Rainbow Turán number of even cycles, repeated patterns and blow-ups of cycles

Cited by 17 publications

References 19 publications

Lower bounds on the Erdős–Gyárfás problem via color energy graphs

Lower bounds on the Erdős–Gyárfás problem via color energy graphs

Rainbow cycles in properly edge-colored graphs

On the Turán number of the hypercube

Contact Info

Product

Resources

About