Integer colorings with forbidden rainbow sums

For a set of positive integers $A \subseteq [n]$, an $r$-coloring of $A$ is rainbow sum-free if it contains no rainbow Schur triple. In this paper we initiate the study of the rainbow Erd\H{o}s-Rothchild problem in the context of sum-free sets, which asks for the subsets of $[n]$ with the maximum number of rainbow sum-free $r$-colorings. We show that for $r=3$, the interval $[n]$ is optimal, while for $r\geq8$, the set $[\lfloor n/2 \rfloor, n]$ is optimal. We also prove a stability theorem for $r\geq4$. The proofs rely on the hypergraph container method, and some ad-hoc stability analysis.Comment: 20 page

Transversals via regularity

Given graphs $G_1,\ldots,G_s$ all on the same vertex set and a graph $H$ with $e(H) \leq s$, a copy of $H$ is transversal or rainbow if it contains at most one edge from each $G_c$. When $s=e(H)$, such a copy contains exactly one edge from each $G_i$. We study the case when $H$ is spanning and explore how the regularity blow-up method, that has been so successful in the uncoloured setting, can be used to find transversals. We provide the analogues of the tools required to apply this method in the transversal setting. Our main result is a blow-up lemma for transversals that applies to separable bounded degree graphs $H$. Our proofs use weak regularity in the $3$-uniform hypergraph whose edges are those $xyc$ where $xy$ is an edge in the graph $G_c$. We apply our lemma to give a large class of spanning $3$-uniform linear hypergraphs $H$ such that any sufficiently large uniformly dense $n$-vertex $3$-uniform hypergraph with minimum vertex degree $\Omega(n^2)$ contains $H$ as a subhypergraph. This extends work of Lenz, Mubayi and Mycroft

Pay It Forward: Unraveling the Role of Cause-related Marketing in the Curvilinear Relationship between Price-oriented Function Usage and Consumer Satisfaction

Price-oriented functions have been prevalently used by sellers for attracting consumers on e-marketplace platforms. However, existing literature has mixed understandings about its influence on improving consumer satisfaction. Besides, few studies have considered how cause-related marketing moderates the impact of price-oriented function usage. Therefore, this paper firstly explores the curvilinear relationship between price-oriented function usage and consumer satisfaction by adopting the repertoire perspective, then further considers the moderating role of cause-related marketing. This study collected data on 29,506 products from one e-marketplace platform in China. By using fixed-effects regression models, it is found that price-oriented function usage (i.e., volume and heterogeneity) have inverted U-shaped relationships with consumer satisfaction. In addition, cause-related marketing weakens the impact of price-oriented function usage heterogeneity on consumer satisfaction. This study contributes to research about platform function usage and guides sellers in terms of using those functions to stimulate consumer satisfaction

Towards Generalizable Deepfake Detection by Primary Region Regularization

The existing deepfake detection methods have reached a bottleneck in generalizing to unseen forgeries and manipulation approaches. Based on the observation that the deepfake detectors exhibit a preference for overfitting the specific primary regions in input, this paper enhances the generalization capability from a novel regularization perspective. This can be simply achieved by augmenting the images through primary region removal, thereby preventing the detector from over-relying on data bias. Our method consists of two stages, namely the static localization for primary region maps, as well as the dynamic exploitation of primary region masks. The proposed method can be seamlessly integrated into different backbones without affecting their inference efficiency. We conduct extensive experiments over three widely used deepfake datasets - DFDC, DF-1.0, and Celeb-DF with five backbones. Our method demonstrates an average performance improvement of 6% across different backbones and performs competitively with several state-of-the-art baselines.Comment: 12 pages. Code and Dataset: https://github.com/xaCheng1996/PRL

Rainbow Hamilton cycle in hypergraph systems

R\"{o}dl, Ruci\'{n}ski and Szemer\'{e}di proved that every $n$-vertex $k$-graph $H$, $k\geq3, \gamma>0$ and $n$ is sufficiently large, with $\delta_{k-1}(H)\geq(1/2+\gamma)n$ contains a tight Hamilton cycle, which can be seen as a generalization of Dirac's theorem in hypergraphs. In this paper, we extend this result to the rainbow setting as follows. A $k$-graph system $\textbf{H}=\{H_i\}_{i\in[m]}$ is a family of not necessarily distinct $k$-graphs on the same $n$-vertex set $V$, a $k$-graph $G$ on $V$ is rainbow if $E(G)\subseteq\bigcup_{i\in[m]}E(H_i)$ and $|E(G)\cap E(H_i)|\leq 1$ for $i\in[m]$. Then we show that given $k\geq3, \gamma>0$, sufficiently large $n$ and an $n$-vertex $k$-graph system $\textbf{H}=\{H_i\}_{i\in[n]}$, if $\delta_{k-1}(H_i)\geq(1/2+\gamma)n$ for $i\in[n]$, then there exists a rainbow tight Hamilton cycle.Comment: 20 pages,5 figure

Influence of the Anteromedial Portal and Transtibial Drilling Technique on Femoral Tunnel Lengths in ACL Reconstruction: Results Using an MRI-Based Model

BACKGROUND In anatomic anterior cruciate ligament (ACL) reconstruction, graft placement through the anteromedial (AM) portal technique requires more horizontal drilling of the femoral tunnel as compared with the transtibial (TT) technique, which may lead to a shorter femoral tunnel and affect graft-to-bone healing. The effect of coronal and sagittal femoral tunnel obliquity angle on femoral tunnel length has not been investigated. PURPOSE To compare the length of the femoral tunnels created with the TT technique versus the AM portal technique at different coronal and sagittal obliquity angles using the native femoral ACL center as the starting point of the femoral tunnel. The authors also assessed sex-based differences in tunnel lengths. STUDY DESIGN Descriptive laboratory study. METHODS Magnetic resonance imaging scans of 95 knees with an ACL rupture (55 men, 40 women; mean age, 26 years [range, 16-45 years]) were used to create 3-dimensional models of the femur. The femoral tunnel was simulated on each model using the TT and AM portal techniques; for the latter, several coronal and sagittal obliquity angles were simulated (coronal, 30°, 45°, and 60°; sagittal, 45° and 60°), representing the 10:00, 10:30, and 11:00 clockface positions for the right knee. The length of the femoral tunnel was compared between the techniques and between male and female patients. RESULTS The mean ± SD femoral tunnel length with the TT technique was 40.0 ± 6.8 mm. A significantly shorter tunnel was created with the AM portal technique at 30° coronal/45° sagittal (35.5 ± 3.8 mm), whereas a longer tunnel was created at 60° coronal/60° sagittal (53.3 ± 5.3 mm; P < .05 for both). The femoral tunnel created with the AM portal technique at 45° coronal/45° sagittal (40.7 ± 4.8 mm) created a similar tunnel length as the TT technique. For all techniques, the femoral tunnel was significantly shorter in female patients than male patients. CONCLUSION The coronal and sagittal obliquity angles of the femoral tunnel in ACL reconstruction can significantly affect its length. The femoral tunnel created with the AM portal technique at 45° coronal/45° sagittal was similar to that created with the TT technique. CLINICAL RELEVANCE Surgeons should be aware of the femoral tunnel shortening with lower coronal obliquity angles, especially in female patients