Gaius
Gaius is the name of more than one figure in the New Testament. The strands available in UPDV come from three letters: a Corinthian whose house received Paul and the whole congregation, one of the very few Corinthians Paul baptized personally, and the named recipient of the elder's third letter. The name reappears at greetings, at baptism-counts, and at letter-headings — never in narrative — so what UPDV gives is a portrait sketched entirely in epistolary registers.
Gaius the Host at Corinth
Writing from Corinth at the close of Romans, Paul folds Gaius into the chain of personal greetings sent on to Rome. The greeting from Tertius the amanuensis ("I Tertius, who write the letter, greet you⁺ in the Lord," Ro 16:22) and from Timothy and Paul's kinsmen (Ro 16:21) sets the register: named coworkers around Paul at the point of writing. In that company Gaius is identified by hospitality:
"Gaius my host, and of the whole church, greets you⁺. Erastus the treasurer of the city greets you⁺, and Quartus the brother." (Ro 16:23)
The phrase is twofold. He is Paul's own host — Paul is lodging with him as the letter is dictated — and he is host "of the whole church," so the same house that receives the apostle as a guest receives the assembly as its meeting place. Hospitality is the trait that fixes him, exercised at both a private and a congregational scale. He stands in Ro 16:23 alongside Erastus, named as treasurer of the city, and Quartus, named simply "the brother."
Gaius Baptized by Paul
In the opening of 1 Corinthians, Paul confronts factional partisanship at Corinth — slogans of "I am of Paul," "I am of Apollos" — and counts up, almost reluctantly, the few Corinthians he had baptized in person. Gaius is named in that short list:
"I thank God that I baptized none of you⁺, except Crispus and Gaius" (1Co 1:14).
He is paired with Crispus in a two-name exception, his identity fixed by a direct apostolic baptism. The motive Paul gives is anti-factional: "lest any man should say that you⁺ were baptized into my name" (1Co 1:15). Paul then remembers a third — "I baptized also the household of Stephanas: besides, I don't know whether I baptized any other" (1Co 1:16) — and resolves the whole question into vocation: "For Christ didn't send me to baptize, but to preach the good news" (1Co 1:17). Gaius's position in this list is exhibitive, not commendatory: he happens to be one of the few in Corinth who could not later be claimed by any party as a Paul-initiate.
The Corinthian setting and the householder-of-the-whole-church description in Ro 16:23 are written from the same city, so the two passages plausibly hold the same Gaius — a Corinthian baptized early by Paul whose house later hosts him.
Gaius the Beloved
The third letter of John opens by naming its addressee:
"The elder to Gaius the beloved, whom I love in truth." (3Jn 1:1)
The sender-title is "the elder," the addressee is named Gaius, the epithet is "the beloved," and the love-clause pegs the affection to truth as its sphere. The body of the short letter then characterizes the Gaius it is written to. He is one whose soul prospers — the elder's prayer for him is that his outward state and health may catch up: "Beloved, I pray that in all things you may prosper and be in health, even as your soul prospers" (3Jn 1:2). He walks in truth, and brothers had come and borne witness to that walk before the elder (3Jn 1:3); the elder's joy is calibrated to "my children walking in truth" (3Jn 1:4).
What the letter then asks of him is hospitality, framed as a continuation of work he is already doing:
"Beloved, you do a faithful work in whatever you do toward those who are brothers and strangers as well; who bore witness to your love before the church: whom you will do well to set forward on their journey worthily of God." (3Jn 1:5-6)
The traveling brothers had borne witness to his love "before the church"; the elder wants them sent forward "worthily of God" because "for the sake of the Name they went forth, taking nothing of the Gentiles" (3Jn 1:7). The conclusion ties the whole back to truth: "We therefore ought to welcome such, that we may be coworkers for the truth" (3Jn 1:8).
So the third Gaius — whether or not the same man as the Corinthian — is exhibited at exactly the point the Corinthian was: receiving brothers, his house counted as the church's hospitality. The trait the New Testament uniformly attaches to a man named Gaius is the open door.