Somerset 308 (Davies 109, Bess 52*) and 8 for 0 need a further 250 to beat Warwickshire 419 and 146 (Yates 53, Abell 4-39)
Somerset need precisely 250 runs on the final day here to stay neck and neck with Essex in a race for the County Championship that might yet rival the excitement of three years ago when Middlesex so memorably pipped Yorkshire at Lord's and left Somerset heartbroken runners-up.
Essex, winners of an extraordinary contest at Canterbury, have the upper hand, but they are due at Taunton in the final match of the season and with only four rounds remaining the chances of the outcome determining the title must be quite high.
Much may depend, though, on what happens here. Somerset, having trailed by 111 on first innings, have given themselves a clear opportunity by dismissing Warwickshire for 146 but chasing runs on the last day at Edgbaston is seldom straightforward.
Jeetan Patel and three tall seam bowlers will see to that. This Somerset side is packed with batting talent, yet it would be a surprise if this were not one of their trickier assignments and an absorbing last day beckons.
Day three, with the balance tipping one way and then the other, was engaging enough. At the start, needing to add 11 to his overnight score, Steven Davies completed a fine hundred when he drove Oliver Hannon-Dalby to the cover boundary off the last ball of the eighth over of the morning, his 15th four.
He celebrated by lofting Jeetan Patel over long-on for his first six but the Warwickshire offspinner had his revenge the very next ball, which turned and bounced and took the edge to be caught by Will Rhodes at slip, a good reaction catch given that wicketkeeper Tim Ambrose flashed his gloves right across Rhodes's eyeline and may even have nudged the ball on its way.
Somerset were still 77 away from avoiding the follow-on at that moment and when Roelof van der Merwe perished via a loose drive after scoring only a single, Warwickshire could claim to have the upper hand. Yet they missed an opportunity to strengthen their grip before lunch by dismissing Dom Bess.
Facing a ball from Henry Brookes that jagged away from him as he shaped to play to leg on 29, Bess was fortunate that Ambrose did not quite have the reach to make a diving catch after the ball took the edge.
Had it stuck, Somerset would have been eight down for 229. Instead, Bess and Craig Overton were able to extend a valuable partnership eventually worth 65 by 38 runs either side of the interval, picking up a second batting point along the way, before Overton was leg before to Hannon-Dalby with the new ball for 36.
Brother Jamie then chipped in with more useful runs, helping Bess complete a 133-ball half-century, pass the follow-on mark, and secure a third bonus point, shortly before the innings ended with Jamie Overton caught at gully and Jack Brooks at slip in quick succession.
Nonetheless, Warwickshire took a handy lead of 111 into the second half of the match along with seven bonus points, which gave them the confidence of knowing that, whatever the outcome, the gap between themselves and bottom-of-the-table Nottinghamshire was likely to be larger at the end of this round of matches than it was at the beginning.
Somerset, though - more so after learning that Kent had been dismissed for 40 by Essex at Canterbury - had thoughts only of carving out a position from which they could win.
Dismissing the much-touted Dom Sibley cheaply for the second time in the match, leg before to Jack Brooks trying to work to leg, they had early encouragement as Warwickshire began their second innings midway through the afternoon and two wickets in two balls by Tom Abell shortly before tea set them up nicely for the final session.
A loose shot by Rhodes sliced the ball to slip, where James Hildreth plucked it out of the air high to his left, then Sam Hain, the other batsman Warwickshire might have fancied to help them into an ascendant position ahead of the final day, stepped into his first ball to be lbw.
Those two scalps sent Somerset back to their dressing room with confidence brimming and it was not long into the last session when Brooks struck another blow, having Adam Hose caught behind.
Rob Yates, after his maiden century in the first innings, further consolidated the good impression he has been making since his debut in May by passing 50 for the second time in the match.
Yet the life he was given on 49 when Craig Overton, diving to his left at second slip, could not quite hold on to a catch was only a foretaste of what was to come in a sharp spell from his brother Jamie that saw Warwickshire slip from 107 for 4 to 113 for 7.
Striking three times in 12 deliveries, Jamie made sure the chance his brother let go cost only four runs, trapping the left-handed Yates leg before both Michael Burgess and Brookes paid the price for a couple of awful swipes at balls wide of off stump.
For all that Somerset were so full of purpose, it was turning into a pretty feeble innings by Warwickshire, who faded quickly as Abell followed his earlier two in two by removing Patel and Hannon-Dalby for career-best figures of 4 for 39.
Bad light meant that Somerset faced only 17 balls instead of the seven overs still scheduled to be played, yet Abell and Davies did not look too eager to take on Hannon-Dalby and Brookes in the half-light. With no wickets lost, the match is nicely poised.