Many studies have shown that collaborative tasks in immersive virtual reality (VR) and augmented reality (AR) environments can enhance collaborative outcomes in learning, training, and game-based contexts. However, the literature offers limited information on how these environments differentially shape social presence and collaborative engagement, particularly among unfamiliar users. This study addresses this gap by examining differences in social presence and collaborative engagement between VR and AR environments for unfamiliar pairs. A between-subjects experiment used an escape room game that featured three collaborative tasks, identically implemented in both VR and AR. The key difference was that the VR experience was stationary, while the AR experience required physical movement between rooms. The study involved 52 participants, divided into VR and AR groups, where two unfamiliar participants were paired into teams to complete the tasks. The results indicate significant differences in collaborative dynamics and user perception between the two environments. Specifically, VR pairs reported a stronger sense of immersion and flow state, whereas AR pairs demonstrated greater contextual awareness and behavioral coordination. Cybersickness measures also differed between conditions; given the locomotion mismatch, this pattern should be interpreted cautiously and not attributed to the environment alone. This finding improves understanding of the impact of immersive environments on collaborative processes and offers insights for designing collaborative XR applications (e.g., training and game-based teamwork), particularly for unfamiliar users.

Abstract In this paper, we characterize the possible cofinalities of the least $\lambda $ -strongly compact cardinal. We show that, on the one hand, for any regular cardinal, $\delta $ , that carries a $\lambda $ -complete uniform ultrafilter, it is consistent, relative to the existence of a supercompact cardinal above $\delta $ , that the least $\lambda $ -strongly compact cardinal has cofinality $\delta $ . On the other hand, provably the cofinality of the least $\lambda $ -strongly compact cardinal always carries a $\lambda $ -complete uniform ultrafilter.

Abstract For a set x, let ${\cal S}\left( x \right)$ be the set of all permutations of x. We prove in ZF (without the axiom of choice) several results concerning this notion, among which are the following: (1) For all sets x such that ${\cal S}\left( x \right)$ is Dedekind infinite, $\left| {{{\cal S}_{{\rm{fin}}}}\left( x \right)} \right| < \left| {{\cal S}\left( x \right)} \right|$ and there are no finite-to-one functions from ${\cal S}\left( x \right)$ into ${{\cal S}_{{\rm{fin}}}}\left( x \right)$ , where ${{\cal S}_{{\rm{fin}}}}\left( x \right)$ denotes the set of all permutations of x which move only finitely many elements. (2) For all sets x such that ${\cal S}\left( x \right)$ is Dedekind infinite, $\left| {{\rm{seq}}\left( x \right)} \right| < \left| {{\cal S}\left( x \right)} \right|$ and there are no finite-to-one functions from ${\cal S}\left( x \right)$ into seq (x), where seq (x) denotes the set of all finite sequences of elements of x. (3) For all infinite sets x such that there exists a permutation of x without fixed points, there are no finite-to-one functions from ${\cal S}\left( x \right)$ into x. (4) For all sets x, $|{[x]^2}| < \left| {{\cal S}\left( x \right)} \right|$ .

Abstract For a set x, let ${\cal S}\left( x \right)$ be the set of all permutations of x. We prove by the method of permutation models that the following statements are consistent with ZF: (1) There is an infinite set x such that $|\wp \left( x \right)| < |{\cal S}\left( x \right)| < |se{q^{1 - 1}}\left( x \right)| < |seq\left( x \right)|$ , where $\wp \left( x \right)$ is the power set of x, seq (x) is the set of all finite sequences of elements of x, and seq1-1 (x) is the set of all finite sequences of elements of x without repetition. (2) There is a Dedekind infinite set x such that $|{\cal S}\left( x \right)| < |{[x]^3}|$ and such that there exists a surjection from x onto ${\cal S}\left( x \right)$ . (3) There is an infinite set x such that there is a finite-to-one function from ${\cal S}\left( x \right)$ into x.